Answer:
The expected value is 0.87.
Step-by-step explanation:
a) To calculate the expected value X we will first see the posible outcomes. So could take value of 0,1,2,3. We will calculate the probability of each outcome. To do so, we will introduce the following notation. Consider the following tuple (A,B,C) where A is the number of butterflies found by Alice, B the number found for by Bob and C the number found by C. To calculate the probability of the tuple (A,B,C) we will do as follows. If the entry of the tuple is 1, then we will multiply by the probability of the person that found the butterfly. So, if A =1, we will multiply by 0.17(Alice finds a butterfly with probability 0.17). On the other side, if the entry of the tuple is 0, we will multiply by (1-p) where p is the probability of the person that found the butterly. So, if A=0, we will multiply by 0.83. So, for example, consider the tuple (1,0,1). The probability of having this result is 0.17*0.75*0.45 (Alice and Charlotte found a butterfly, but Bob didn't). We can do this since we are said that their probabilities of success don't affect others' probabilities.
We will see the total number of butterflies and the tuples associated to that number. That is
X number of butterflies - tuples
0 butterflies - (0,0,0)
1 butterfly - (1,0,0) or (0,1,0) or (0,0,1)
2 butterflies - (1,1,0) or (1,0,1) or (0,1,1)
3 butterflies - (1,1,1)
To find the probability of the value of X, we will sum up the probability of the associated tuples. The values of the probabilities are as follows
(0, 0, 0) = 0.342375
(0, 0, 1 ) = 0.280125
(0, 1, 0) = 0.114125
(0, 1, 1 ) = 0.093375
(1, 0, 0) = 0.070125
(1, 0, 1 ) = 0.057375 = 0.17*0.75*0.45
(1, 1, 0) = 0.023375
(1, 1, 1) = 0.019125
In this case,
P(X=0) = 0.342375 ,
P(X=1) = 0.464375 = 0.280125 +0.114125+ 0.070125
P(X=2) = 0.174125
P(X =3 ) = 0.019125
So, the expected value of X is given by
0* 0.342375 +1 * 0.464375 +2* 0.174125+3*0.019125 = 0.87
b)Let X1 be the number of butterflies found by Alice, X2 the number found by Bob and X3 the number found by Charlotte. Then X = X1+X2+X3. Using the expected value properties and the independence of X1, X2 and X3 we have that E(X) = E(X1)+E(X2)+E(X3).
Recall that each variable is as follows. Xi is equal to 1 with probability p and it is 0 with probability (1-p). Then, the expected value of Xi is
[tex]1\cdot p + 0\codt (1-p)=p[/tex]. Note that the value of p for X1,X2 and X3 is 17%, 25% and 45% respectively.
Then E(X) = 17%+25%+45%= 0.87.
So the expected number of butterflies is 0.87.
Final answer:
The expected value of the total number of butterflies, X, that Alice, Bob, and Charlotte catch is 0.87. This is found by summing their independent probabilities of catching a butterfly (0.17 for Alice, 0.25 for Bob, and 0.45 for Charlotte). X is also represented as the sum of three indicator random variables X1, X2, and X3, leading to the same expected value.
Explanation:
Expected Value of the Number of Butterflies Caught
In this scenario with Alice, Bob, and Charlotte searching for butterflies in separate parts of a field, the random variable X represents the total number of butterflies they catch. The expected value of X, or E(X), is calculated by adding the individual probabilities of finding a butterfly, since their probabilities are independent.
To find the expected value of X:
Multiply the probability of each person finding a butterfly by the number of butterflies they would find in that event (which is 1 since each either finds 1 butterfly or none), and
Add these products together.
The expected value is thus 0.17 + 0.25 + 0.45 = 0.87 butterflies. We can also express X as X1 + X2 + X3, where each Xi is an indicator random variable for whether Alice (X1), Bob (X2), or Charlotte (X3) found a butterfly.
The expected value for each indicator variable is the same as the person's probability of success. So, E(X1) = 0.17, E(X2) = 0.25, and E(X3) = 0.45. By the linearity of expectation, E(X) = E(X1) + E(X2) + E(X3), which also equals 0.87 butterflies.
Write the values of each of the digits in 6,582,390,417 in word form
The solution is: 6,582,390,417 in word form is, six billon five hundred eighty two million three hundred thousand and ninety four hundred seventeen.
What is place value?Place value is the basis of our entire number system. This is the system in which the position of a digit in a number determines its value.
The number 42,316 is different from 61,432 because the digits are in different positions.
here, we have,
given that,
6,582,390,417
so, the values of each of the digits in 6,582,390,417 in word form is:
6: six billon
5: five hundred
8: eighty two million
3: three hundred thousand
and ninety four hundred
seventeen.
Hence, The solution is: 6,582,390,417 in word form is, six billon five hundred eighty two million three hundred thousand and ninety four hundred seventeen.
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In order to estimate the average time spent on the computer terminals per student at a local university, data were collected for a sample of 81 business students over a one-week period. Assume the population standard deviation is 1.8 hours. Answer questions 7 - 9.
7. What is the standard error of the mean?
a. 7.50
b . 0.39
c. 2.00
d. 0.20
8. With a 0.95 probability, the margin of error is approximately
a. 0.39
b 1.96
c. 0.20
d. 1.64
9. If the sample mean is 9 hours, then the 95% confidence interval is
a. 7.04 to 10.96 hours
b. 7.36 to 10.64 hours
c. 7.80 to 10.20 hours
1d. 8.61 to 9.39 hours
Answer:
7) d)
standard error of the mean of one sample of 'n' observation = 0.20
8) a)
The margin of Error = 0.392
9) d
The 95% of confidence intervals are (8.61 , 9.39)
Step-by-step explanation:
7)
solution:-
The Given data sample size 'n' = 81
Given Population standard deviation 'σ' = 1.8 hours
The standard error of the mean of one sample of 'n' observation is
Standard error (SE)
= [tex]\frac{S.D}{\sqrt{n} }[/tex]
= σ / √n
= [tex]\frac{1.8}{\sqrt{81} } =0.2[/tex]
standard error of the mean of one sample of 'n' observation = 0.20
8)
Solution:-
The Given data sample size 'n' = 81
Given Population standard deviation 'σ' = 1.8 hours
Given the probability is 0.95
The z- score = 1.96 at 0.05 level of significance.
The margin of Error = [tex]\frac{z_{0.95} S.D}{\sqrt{n} }[/tex]
= [tex]\frac{1.96 (S.D)}{\sqrt{n} }[/tex]
= [tex]\frac{1.96 (1.8)}{\sqrt{81} }[/tex]
= 0.392
The margin of Error = 0.392
9)
Solution:-
The 95% of confidence intervals are
[tex](x^{-} - 1.96\frac{S.D}{\sqrt{n} } , x^{-} + 1.96\frac{S.D}{\sqrt{n} } )[/tex]
[tex](9 - 1.96\frac{1.8}{\sqrt{81} } , 9+ 1.96\frac{1.8}{\sqrt{81} } )[/tex]
(9 - 0.392 , (9 + 0.392)
(8.609 , 9.392)
The 95% of confidence intervals are (8.61 , 9.39)
According to the U.S. Bureau of the Census, about 75% of commuters in the United States drive to work alone. Suppose 150 U.S. commuters are randomly sampled. (a) What is the probability that fewer than 101 commuters drive to work alone
Answer:
1.19% probability that fewer than 101 commuters drive to work alone
Step-by-step explanation:
I am going to use the binomial approximation to the normal to solve this question.
Binomial probability distribution
Probability of exactly x sucesses on n repeated trials, with p probability.
Can be approximated to a normal distribution, using the expected value and the standard deviation.
The expected value of the binomial distribution is:
[tex]E(X) = np[/tex]
The standard deviation of the binomial distribution is:
[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]
Normal probability distribution
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
When we are approximating a binomial distribution to a normal one, we have that [tex]\mu = E(X)[/tex], [tex]\sigma = \sqrt{V(X)}[/tex].
In this problem, we have that:
[tex]p = 0.75, n = 150[/tex]
[tex]\mu = E(X) = 150*0.75 = 112.5[/tex]
[tex]\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{150*0.75*0.25} = 5.3[/tex]
(a) What is the probability that fewer than 101 commuters drive to work alone
Using continuity corretion, this is [tex]P(X < 101-0.5) = P(X < 100.5)[/tex], which is the pvalue of Z when X = 100.5. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{100.5 - 112.5}{5.3}[/tex]
[tex]Z = -2.26[/tex]
[tex]Z = -2.26[/tex] has a pvalue of 0.0119
1.19% probability that fewer than 101 commuters drive to work alone
The probability that fewer than 101 U.S. commuters drive to work alone, based on a 75% solo driving rate, is calculated using the binomial probability formula, resulting in the answer.
To solve this problem, we can use the binomial probability formula, as this is a binomial distribution (success/failure) with a known probability of success.
The formula for the probability mass function of a binomial distribution is:
[tex]\[ P(X = k) = \binom{n}{k} \cdot p^k \cdot (1 - p)^{n - k} \][/tex]
where:
-[tex]\( n \)[/tex] is the number of trials (sample size),
- [tex]\( k \)[/tex] is the number of successful outcomes,
- [tex]\( p \)[/tex]is the probability of success on a single trial.
In this case, [tex]\( n = 150 \)[/tex] (number of commuters),[tex]\( p = 0.75 \)[/tex] (probability of driving alone), and we want to find the probability that fewer than 101 commuters drive alone (\( k < 101 \)).
[tex]\[ P(X < 101) = P(X \leq 100) = \sum_{k=0}^{100} \binom{150}{k} \cdot 0.75^k \cdot (1 - 0.75)^{150 - k} \][/tex]
Now, we can use a calculator or statistical software to compute this probability. Keep in mind that the binomial coefficient[tex]\(\binom{n}{k}\)[/tex] is the number of ways to choose \(k\) successes from \(n\) trials and can be calculated as[tex]\(\frac{n!}{k! \cdot (n - k)!}\).[/tex]
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A map shows the straight-line distance from George’s house to his school as 9.5 centimeters. If George lives 475 meters from his school, what is the scale of the map?
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Answer:
1/5000
Step-by-step explanation:
The scale is ...
map distance / ground distance = (0.095 m)(475 m) = 0.0002 = 1/5000
To find the map scale, convert the actual distance from meters to centimeters and then set up a ratio with the map distance. After simplifying, the scale of the map is determined to be 1:5000, meaning 1 centimeter on the map equals 5000 centimeters in reality.
To determine the scale of the map we'll convert the given distance from George's house to the school into the same units and then create a ratio. Since the map shows the distance as 9.5 centimeters and the actual distance is 475 meters, we first need to convert meters to centimeters because the scale needs to have the same units for both measurements.
1 meter = 100 centimeters, so 475 meters is equivalent to 475 x 100 = 47500 centimeters.
Now, we can set up the scale of the map as a ratio:
9.5 centimeters (map distance) / 47500 centimeters (actual distance).
We can simplify this ratio to find the scale of the map by dividing both the numerator and the denominator by 9.5:
9.5 cm / 47500 cm = 1 cm / (47500 / 9.5) = 1 cm / 5000 cm
This means that every centimeter on the map represents 5000 centimeters in real life. Therefore, the map scale is 1:5000.
25. Michael is studying population changes in two
types of birds living on an island. Compare the
populations by finding and interpreting the
average rates of change over the interval [0, 18]
Bird A
Time(months):
6
12
18
Population:
(thousands)
8.3
8.6
8.8
9.1
Bird B
y= 3.6(1.06)x
Final answer:
To compare the populations of the two bird species, we calculate the average rate of change for Bird A by dividing the change in population by time over 18 months, and for Bird B by evaluating the given exponential function at the endpoints of the time interval.
Explanation:
To compare the population changes of Bird A and Bird B and interpret the average rates of change over the interval [0, 18], we first need to calculate the average rate of change for Bird A. Given Bird A's population at different times, we can calculate the average rate of change by dividing the change in population by the change in time, over the interval [0, 18].
For Bird A, the population increases from 8.3 to 9.1 thousand over 18 months. The average rate of change for Bird A is thus (9.1 - 8.3) / (18 - 0) = 0.8 / 18 = 0.0444 thousand per month.
For Bird B, the population change is given by a function y= 3.6(1.06)ˣ, where y is the population in thousands and x is the time in months. To find the average rate of change over [0, 18], we evaluate the function at the endpoints of the interval: y(0) = 3.6 and y(18) = 3.6(1.06)¹⁸. After calculating y(18), we'd use the same average rate of change formula.
Interpreting the results, if Bird A's average rate of change is less than that of Bird B, it means Bird B's population is growing faster on average than Bird A's population over the 18 months.
What does the 99% confidence level in the previous problem tell us? Group of answer choices There is a 99% chance that this particular interval contains the population proportion. The interval will contain 99% of all sample proportions. Of confidence intervals with this margin of error, 99% will contain the population proportion. Of confidence intervals with this margin of error, 99% will contain the sample proportion.
Answer:
The confidence interval for the mean is given by the following formula:
[tex]\hat p \pm z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}[/tex]
The margin of error is given by:
[tex] ME= z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}[/tex]
For this case since the confidence is 99% we are confident that the true proportion of interest would be on the interval calculated and the best option for this case is:
Of confidence intervals with this margin of error, 99% will contain the population proportion
Step-by-step explanation:
Previous concepts
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
Solution to the problem
The confidence interval for the mean is given by the following formula:
[tex]\hat p \pm z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}[/tex]
The margin of error is given by:
[tex] ME= z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}[/tex]
For this case since the confidence is 99% we are confident that the true proportion of interest would be on the interval calculated and the best option for this case is:
Of confidence intervals with this margin of error, 99% will contain the population proportion
A sailboat is sitting at rest near its dock. A rope attached to the bow of the boat is drawn in over a pulley that stands on a post on the end of the dock that is 5 feet higher than the bow. If the rope is being pulled in at a rate of 2 feet per second, how fast is the boat approaching the dock when the length of rope from bow to pulley is 13 feet
Answer:
The boat is approaching the dock at rate of 2.14 ft/s
Step-by-step explanation:
The situation given in the question can be modeled as a triangle, please refer to the attached diagram.
A rope attached to the bow of the boat is drawn in over a pulley that stands on a post on the end of the dock that is 5 feet higher than the bow that means x = 5 ft.
The length of rope from bow to pulley is 13 feet that means y = 13 ft.
We know that Pythagorean theorem is given by
[tex]x^{2} + y^{2} = z^{2}[/tex]
Differentiating the above equation with respect to time yields,
[tex]2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 2z\frac{dz}{dt}[/tex]
[tex]x\frac{dx}{dt} + y\frac{dy}{dt} = z\frac{dz}{dt}[/tex]
dx/dt = 0 since dock height doesn't change
[tex]y\frac{dy}{dt} = z\frac{dz}{dt}[/tex]
[tex]\frac{dy}{dt} = \frac{z}{y} \frac{dz}{dt}[/tex]
The rope is being pulled in at a rate of 2 feet per second that is dz/dt = 2 ft/s
First we need to find z
z² = (5)² + (13)²
z² = 194
z = √194
z = 13.93 ft
So,
[tex]\frac{dy}{dt} = \frac{z}{y} \frac{dz}{dt}[/tex]
[tex]\frac{dy}{dt} = \frac{13.93}{13}(2)[/tex]
[tex]\frac{dy}{dt} = 2.14[/tex] [tex]ft/s[/tex]
Therefore, the boat is approaching the dock at rate of 2.14 ft/s
So, the boat approached the dock with a speed of 2.1337 m/sec.
Pythagoras Theorem:Pythagoras Theorem is an important topic in Maths, which explains the relation between the sides of a right-angled triangle.
So, the formula is,
[tex]a^2+b^2=c^2[/tex]
Differentiating the above equation,
[tex]2a\frac{da}{dt} +2b\frac{db}{dt} =2c\frac{dc}{dt} ...(1)[/tex]
It is given that,
[tex]a=5m\\\frac{da}{dt}=0\\ c=13\\\frac{dc}{dt}=2 m/s[/tex]
[tex]b=\sqrt{13^2-5^2} \\b=12 m/s[/tex]
Substituting the above values in equation (1) we get,
[tex]2\times5\times0+2\times12\frac{db}{dt} =2\times13\times2\\\frac{db}{dt} =\frac{26}{12}\\ \frac{db}{dt} =2.1337 m/s[/tex]
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The joint pdf of pressures for right and left front tires is given in Exercise 9. a. Determine the conditional pdf of Y given that X 5 x and the conditional pdf of X given that Y 5 y. b. If the pressure in the right tire is found to be 22 psi, what is the probability that the left tire has a pressure of at least 25 psi? Compare this to P(Y $ 25).
Answer:
Step-by-step explanation:
the explanation is attached below
The conditional pdfs of Y given that X is 5 x and X given that Y is 5 y is 20< y<30.
Explain about the probability?
Calculating or estimating how likely something is to occur is what probability is all about. The likelihood of an event occurring can be expressed using words like "certain," "impossible," or "probable." Probabilities are always expressed in mathematics as fractions, decimals, or percentages with values ranging from 0 to 1.
The definition of probability, methods for calculating the probabilities of single and multiple random events, and the distinction between probabilities and odds of an event occurring are all covered in this article. Key conclusions: The probability that an event will occur is determined by probability: P(A) = f / N.
f y/x(y/x) = f(x, y)/f x(x)
=K(x²+y²)/ 10Kx²+0.05
0
20<y<30
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f(x) = 2x-1
What is f(13)?
Answer:
f(13)=2x-1, x=7
Step-by-step explanation:
7*2=14, 14-1=13
vise versa
f(13)=2(7)-1
Answer: 25
Step-by-step explanation: Easy. Just substitute 13 for x into the equation. f(13)=2(13)-1
= 26-1
= 25
Can someone please help me with this calculus problem? I am very confused on how to complete it.
Divide a 14-in diameter pizza into three slices of the equal area using two parallel cuts.
Approximate approach: use Riemann sums to estimate where to place the cuts
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Answer:
cut the pizza ±1.855 inches from the centerline
Step-by-step explanation:
Here's an interesting approach that actually gives amazing accuracy.
If the pizza were a 14" square, the cuts would be 14/6 = 2.33 inches from the centerline. If we draw a diagram of the pizza as a circle of radius 7 centered at the origin, we propose to create a running total of the areas of trapezoidal slices in the first quadrant parallel to the y-axis out to a distance of x=2.3. We want to find where the accumulated area is close to 1/12 of the area of the circle. In order to get sufficient resolution as to where the cut should be, we choose to use a total of 20 slices to cover that area. (n=20 for the Riemann sum)
Of course, the equation of the circle is x^2 +y^2 = 49. Then the value of y of interest is ...
f(x) = √(49 -x^2)
where the value of x is some multiple of 2.3/20, the slice width for our Riemann sum. If we number the slices 0 to 19, the accumulated area up to slice k is ...
[tex]\displaystyle A_k=\dfrac{2.3}{20}\sum_{n=0}^{k}\dfrac{f(x_n)+f(x_{n+1})}{2}[/tex]
As you can tell, this is using the trapezoidal method of computing the Riemann sum. We considered left, right, and midpoint integration methods but settled on this as having the kind of accuracy we wanted.
We want an accumulated area of 1/3 of our quarter circle, or (1/12)(π)(7^2) = 49π/12, so we want to find the zero of the difference Ak -49π/12.
As it turns out, the desired area is bracketed by k=16 and k=17. By linearly interpolating between the area values for these numbers of slices, we find that x=1.8546 is the location we need to cut the pizza. The problem is symmetrical, so the other cut is at x = -1.8546 inches from center.
_____
When evaluating functions multiple times, it is convenient to use a graphing calculator or spreadsheet. With a spreadsheet, you could list function values in one column, the trapezoidal method area in another column, and the accumulated area in yet another column. As here, the interpolation required for a final answer is also easily handled in a spreadsheet.
__
Additional comment
The arc α that encloses a sector equal to 1/3 the area of the pizza will satisfy the equation α -sin(α) -2π/3 = 0. That is about 2.6053256746 radians. The cut distance we're trying to find is the 7cos(α/2) ≈ 1.85452459222 inches, which differs by about 0.005% from the value we found.
What are numbers that are positive negative and 0 that are not irrational
Answer:
Any number that is not a fractional value positive and negative but technically all numbers are irrational
Step-by-step explanation:
Any number that is not a fractional value positive and negative but technically all numbers are irrational.
We have to find the numbers that are positive negative and 0 that are not irrational
What is the fractional value positive?
Fractional values are represented using fixed-point arithmetic and are useful for DSP applications.
For a fractional division, we first scale the denominator to the range 0.5 ≤ d < 1.0.
Then we use a table lookup to provide an estimate of x0 to d−1.
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Assume the random variable X is normally distributed with mean mu equals 50 and standard deviation sigma equals 7. Compute the probability. Be sure to draw a normal curve with the area corresponding to the probability shaded. Upper P (Upper X greater than 34 )
Answer:
[tex]P(X>34) = 0.9889[/tex]
Step-by-step explanation:
We are given the following information in the question:
Mean, μ = 50
Standard Deviation, σ = 7
We are given that the distribution of random variable X is a bell shaped distribution that is a normal distribution.
Formula:
[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]
P(X greater than 34)
[tex]P( X > 34) = P( z > \displaystyle\frac{34 - 50}{7}) = P(z > -2.2857)[/tex]
[tex]= 1 - P(z \leq -2.2857)[/tex]
Calculation the value from standard normal z table, we have,
[tex]P(X>34) = 1 - 0.0111= 0.9889= 98.89\%[/tex]
The attached image shows the normal curve.
There is a math joke that floats around the internet every once in a while that goes something like . . . A mathematician and her infinitely many friends decide to stop by their favorite bar. The first mathematician orders 1 beer, the second orders 1/2 a beer, the third orders 1/4 a beer, the fourth orders 1/8 a beer, the fifth orders 1/16 a beer. . . it’s at this point the bartender, knowing a little math herself, interrupts and says "Fine, I’ll just pour you two beers.". How many beers should the bartender pour if the first mathematician orders 6 beers, the second orders 2 beer, the third orders 2/3 a beer, and so on?
Answer:
Three beers.Step-by-step explanation:
The first mathematician orders 1 beer, the second orders 1/2 a beer, the third orders 1/4 a beer, the fourth orders 1/8 a beer, the fifth orders 1/16 a beer. . .
The Sequence is: [tex]1,\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16} \cdots[/tex]On observation, the next term is derived through the division of the previous term by 2.2 is the constant factorTherefore, the bartender says 'Fine, I’ll just pour you two beers.'Therefore, if the first mathematician orders 6 beers, the second orders 2 beer, the third orders 2/3 a beer, and so on
The sequence is [tex]6,2,\frac{2}{3}, \cdots[/tex]On observation, the next term is derived through the division of the previous term by 3.3 is the constant factorTherefore, the bartender should pour three beers.To find out how many beers the bartender should pour for the infinite series of mathematicians ordering in a pattern starting with 6 beers and each following ordering two-thirds of the previous, we use the sum formula for a geometric series. The sum is 18 beers.
The student is asking a question related to an infinite series in mathematics. In the scenario described, the first mathematician orders 6 beers, the second orders 2 beers, and each mathematician that follows orders a fraction of the previous mathematician's order, specifically two-thirds of the previous amount. To find out how many beers the bartender should pour, we need to find the sum of the geometric series.
The general form of a geometric series is a + ar + ar² + ar³ + ..., where 'a' is the first term and 'r' is the common ratio between terms. In this case, the first term 'a' is 6 (the first order) and the common ratio 'r' is 2/3 (each subsequent mathematician orders two-thirds of the previous one's amount).
To find the sum of this infinite series, we can use the formula S = a / (1 - r), when |r| < 1. Substituting the values from our question, we get S = 6 / (1 - 2/3) = 6 / (1/3) = 6 * 3 = 18 beers. Therefore, the bartender should pour 18 beers.
Graph the line that passes through the points (3, -2)(3,−2) and (9, 2)(9,2) and determine the equation of the line.
Answer:
2x -3y = 12
Step-by-step explanation:
For some horizontal change Δx and some vertical change Δy between the two points, an equation of the line through points (x1, y1) and (x2, y2) can be written as ...
Δy·x -Δx·y = Δy·(x1) -Δx·(y1)
Here, we have ...
Δy = y2 -y1 = 2 -(-2) = 4
Δx = x2 -x1 = 9 -3 = 6
So, our equation can be ...
4x -6y = 4·3 -6·(-2) = 24
Factoring out a common factor of 2 makes the equation be ...
2x -3y = 12 . . . . . . equation of the line in standard form
Solving for y gives the equation in slope-intercept form:
y = 2/3x -4
_____
More conventional solution
Plotting the points and drawing the line, you see that the y-intercept is -4. You also see that there is a "rise" of 2 grid squares for each "run" of 3 grid squares. Thus the slope of the line is 2/3. With this information, you can write the equation directly in slope-intercept form:
y = mx + b . . . . . . line with slope m and y-intercept b
y = 2/3x -4 . . . . . . the line through the given points
Final answer:
The line passing through the points (3, -2) and (9, 2) has a slope of 2/3, and its equation is y = (2/3)x - 4, which can be graphed by plotting the given points and ensuring the slope is represented correctly.
Explanation:
To graph the line that passes through the points (3, -2) and (9, 2), we first find the slope of the line. The slope formula is (y2 - y1) / (x2 - x1). Plugging in our points, we get (2 - (-2)) / (9 - 3) which simplifies to 4 / 6, further reduced to 2 / 3. Therefore, the slope of the line is 2 / 3.
Next, we use one of the points and the slope to write the equation in point-slope form, y - y1 = m(x - x1). Using the point (3, -2), the equation becomes y + 2 = (2/3)(x - 3). After distributing the slope and moving -2 to the other side, we get the equation y = (2/3)x - 4.
Finally, we can graph the line by plotting the two given points and drawing a straight line through them, ensuring that the rise over the run matches the slope of 2 / 3. The equation of the line y = (2/3)x - 4 can be verified using various x-values to see if the resulting y-values fall on the line plotted.
A box designer has been charged with the task of determining the surface area of various open boxes (no lid) that can be constructed by cutting four equal-sized surface corners from an 8-inch by 11.5 inch sheet of cardboard and folding up the sides.
1. Determine a function that relates the total surface area, s, (measured in square inches) of the open box to the size of the square cutout x (measured in inches).
2. What is the domain and range of the function s?
3. What is the surface area when a 1" x 1" square is cut out?
4. What size square cutout will result in a surface area of 20 in?
5. What is the surface area of the box when the volume is maximized? (Calculator)
Answer:
A box designer has been charged with the task of determining the surface area of various open boxes (no lid) that can be constructed by cutting four equal-sized surface corners from an 8-inch by 11.5 inch sheet of cardboard and folding up the sides.
1. Determine a function that relates the total surface area, s, (measured in square inches) of the open box to the size of the square cutout x (measured in inches).
2. What is the domain and range of the function s?
3. What is the surface area when a 1" x 1" square is cut out?
4. What size square cutout will result in a surface area of 20 in?
5. What is the surface area of the box when the volume is maximized? (Calculator)
Step-by-step explanation:
Banks can create money by making use of:
Answer:
Loans
Step-by-step explanation:
I don´t know how to explain it,and I hope my answer is correct though.
Answer:
banks create money by issuing loans and opening checking accounts
Step-by-step explanation:
Automated manufacturing operations are quite precise but still vary, often with distribution that are close to Normal. The width in inches of slots cut by a milling machine follows approximately the N(0.72,0.0012) distribution. The specifications allow slot widths between 0.71975 and 0.72025. What proportion of slots meet these specifications
Answer:
The proportion of slots which meet these specifications is 0.16634 or 16.63%.
Step-by-step explanation:
We are given that the width in inches of slots cut by a milling machine follows approximately the N(0.72,0.0012) distribution.
Also, the specifications allow slot widths between 0.71975 and 0.72025.
Let X = width in inches of slots cut by a milling machine
The z-score probability distribution for normal distribution is given by;
Z = [tex]\frac{ X-\mu}{{\sigma}} }} }[/tex] ~ N(0,1)
where, [tex]\mu[/tex] = population mean width = 0.72
[tex]\sigma[/tex] = standard deviation = 0.0012
The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X.
Now, Probability that the specifications allow slot widths between 0.71975 and 0.72025 is given by = P(0.71975 < X < 0.72025)
P(0.71975 < X < 0.72025) = P(X < 0.72025) - P(X [tex]\leq[/tex] 0.71975)
P(X < 0.72025) = P( [tex]\frac{ X-\mu}{{\sigma}} }} }[/tex] < [tex]\frac{ 0.72025-0.72}{{0.0012}} }} }[/tex] ) = P(Z < 0.21) = 0.58317
P(X [tex]\leq[/tex] 0.71975) = P( [tex]\frac{ X-\mu}{{\sigma}} }} }[/tex] [tex]\leq[/tex] [tex]\frac{ 0.71975-0.72}{{0.0012}} }} }[/tex] ) = P(Z [tex]\leq[/tex] -0.21) = 1 - P(Z < 0.21)
= 1 - 0.58317 = 0.41683
So, in the z table the P(Z [tex]\leq[/tex] x) or P(Z < x) is given. So, the above probability is calculated by looking at the value of x = 0.21 in the z table which has an area of 0.58317.
Therefore, P(0.71975 < X < 0.72025) = 0.58317 - 0.41683 = 0.16634
Hence, the proportion of slots who meet these specifications is 16.63%.
How to solve -222=5x-3(-7y-4)
Answer:
x=46 4/5, x=46.8
Step-by-step explanation:
To find x-intercept/zero, substitute y=0
-222=5x-3(-7*0-4)
-222=5x-3(-7*-4) Solve
x= -234/5
x=46 4/5, x=46.8
A fence is to be built to enclose a rectangular area of 450450 square feet. The fence along three sides is to be made of material that costs $33 per foot. The material for the fourth side costs $99 per foot. Find the dimensions of the rectangle that will allow for the most economical fence to be built.
Answer:
The dimensions of the rectangle that will allow for the most economical fence to be built are 30x15 feets, where two sides of 30 feets long cost $33 each one per foot, one side of 15 feets costs also $33 and the remaining side costs $99
Step-by-step explanation:
If x and y were the dimensions of the rectangle (in feets), then we have that x*y = 450. Therefore, y = 450/x.
Note that the rectangle as a result is formed by 2 sides with length x and 2 other sides with length 450/x. Lets suppose that x is the length of the 2 sides that costs both $33 and the other two sides, which have length 450/x, one costs also $33 and the other costs $99.
The cost, in $, function f,in terms of x, is given as follows
[tex] f(x) = 2 * 33 * x + 33*\frac{450}{x} + 99*\frac{450}{x} = 66x + \frac{59400}{x} [/tex]
We want to minimize f, so we will derivate it and equalize the derivate to 0:
[tex] f'(x) = 66 - \frac{59400}{x^2} [/tex]
[tex] f'(x) = 0 \leftrightarrow 66 = \frac{59400}{x^2} \leftrightarrow x^2 = \frac{59400}{66} = 900 \leftrightarrow x = \sqrt{900} = 30 [/tex]
(Note that x cant be negative, so in the equation we didnt count the opposite of the square root of 900)
We concluded that one dimension is 30 feets, and the other should be 450/30 = 15.
Suppose that a company wishes to predict sales volume based on the amount of advertising expenditures. The sales manager thinks that sales volume and advertising expenditures are modeled according to the following linear equation. Both sales volume and advertising expenditures are in thousands of dollars. Estimated Sales Volume=46.41+0.45(Advertising Expenditures) If the company has a target sales volume of $200,000, how much should the sales manager allocate for advertising in the budget? Round your answer to the nearest dollar.
Final answer:
The sales manager should allocate approximately $341,310 for advertising to achieve the target sales volume of $200,000. This amount is determined by using the linear equation provided and solving for the advertising expenditures.
Explanation:
To find out how much the sales manager should allocate for advertising expenditures to achieve a target sales volume of $200,000, we use the given linear equation:
Estimated Sales Volume = 46.41 + 0.45(Advertising Expenditures)
First, we convert the target sales volume to thousands of dollars - which would be $200 (since $200,000 is in thousands), and then plug it into the equation:
200 = 46.41 + 0.45(Advertising Expenditures)
Next, we solve for Advertising Expenditures:
200 - 46.41 = 0.45(Advertising Expenditures)
153.59 = 0.45(Advertising Expenditures)
Advertising Expenditures = 153.59 / 0.45
Advertising Expenditures = 341.31
Therefore, the sales manager should allocate approximately $341,310 for advertising in the budget to achieve the target sales volume of $200,000. This value is rounded to the nearest dollar as requested.
To achieve a target sales volume of $200,000, the company should allocate approximately $341,310 for advertising.
To determine the amount to allocate for advertising to reach a target sales volume of $200,000, we use the provided linear equation:
⇒ Estimated Sales Volume = 46.41 + 0.45(Advertising Expenditures)
First, set the estimated sales volume to 200 (since the sales volume is in thousands of dollars):⇒ 200 = 46.41 + 0.45(Advertising Expenditures)
Next, solve for Advertising Expenditures:Subtract 46.41 from both sides:⇒ 200 - 46.41 = 0.45(Advertising Expenditures)
⇒ 153.59 = 0.45(Advertising Expenditures)
Divide both sides by 0.45:⇒ Advertising Expenditures = 153.59 ÷ 0.45
⇒ Advertising Expenditures ≈ 341.31
Therefore, the company should allocate approximately $341,310 (rounded to the nearest dollar) for advertising expenditures to achieve the target sales volume of $200,000.
Complete question:
Suppose that a company wishes to predict sales volume based on the amount of advertising expenditures. The sales manager thinks that sales volume and advertising expenditures are modeled according to the following linear equation. Both sales volume and advertising expenditures are in thousands of dollars.
Estimated Sales Volume = 46.41 + 0.45(Advertising Expenditures)
If the company has a target sales volume of $200,000, how much should the sales manager allocate for advertising in the budget? Round your answer to the nearest dollar.
-7y = -91, what is the value of y
Answer: The value of y is 13
Step-by-step explanation: To find the value of y, we will use properties of equality.
Step 1: -7y = -91 (We want to find the value of y, or 1 y)
Step 2: (Use the division property of equality) -7y/-7 = -91/-7
Step 3: (Answer) y = 13
Answer:
y= 13
Step-by-step explanation:
-7y = -91
divide both sides by -7
y = 13
Use Polya's four-step method in problem solving to solve the following problem. Five runners, Andy, Beth, Dale, Ella, and Tri, are in a one-mile race. Dale finished 5 second(s) before Tri. Tri finished 7 second (s) after beth. beth finished 7 second(s) after Ella. Ella finished 4 deco d(s) before andy. In which order did the runners finish the race?
Andy Beth Tri Dale Ella.
To solve the problem using Polya's four-step method, assign variables to represent the runners, set up a system of equations, and solve for the unknowns.
Explanation:To solve this problem using Polya's four-step method, we need to identify the given information and the unknowns. Let's assign variables to represent the runners: Andy (A), Beth (B), Dale (D), Ella (E), and Tri (T). From the given information, we know that Dale finished 5 seconds before Tri (D - T = 5), Tri finished 7 seconds after Beth (T - B = 7), Beth finished 7 seconds after Ella (B - E = 7), and Ella finished 4 seconds before Andy (E - A = 4). Now, we can set up a system of equations to solve for the order of the runners.
From the equations, we can solve for the values of the variables. Plugging the values back into the original equations, we find that the order in which the runners finished the race is Andy, Ella, Beth, Tri, and Dale.
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The lifetime of a certain type of battery is normally distributed with mean value 11 hours and standard deviation 1 hour. There are nine batteries in a package. What lifetime value is such that the total lifetime of all batteries in a package exceeds that value for only 5% of all packages?
The total lifetime value for 5% of packages to exceed is approximately 103.94 hours.
To solve this problem, we first need to find the distribution of the total lifetime of all nine batteries in a package. Since each battery's lifetime follows a normal distribution with a mean of 11 hours and a standard deviation of 1 hour, the total lifetime of all nine batteries will also follow a normal distribution.
The mean of the total lifetime of all nine batteries is [tex]\( 9 \times 11 = 99 \)[/tex] hours.
The standard deviation of the total lifetime of all nine batteries is [tex]\( \sqrt{9} \times 1 = 3 \)[/tex] hours.
Now, we need to find the value such that the total lifetime of all batteries in a package exceeds that value for only 5% of all packages. This is equivalent to finding the 95th percentile of the distribution of the total lifetime.
We'll use the Z-score formula to find the Z-score corresponding to the 95th percentile, and then use that Z-score to find the corresponding value in terms of hours.
The Z-score corresponding to the 95th percentile is approximately 1.645 (you can find this value from standard normal distribution tables or calculators).
Now, we'll use the formula:
[tex]\[ \text{Value} = \text{Mean} + (\text{Z-score} \times \text{Standard deviation}) \][/tex]
[tex]\[ \text{Value} = 99 + (1.645 \times 3) \][/tex]
[tex]\[ \text{Value} = 99 + 4.935 \][/tex]
[tex]\[ \text{Value} \approx 103.94 \][/tex]
So, the total lifetime value such that the total lifetime of all batteries in a package exceeds that value for only 5% of all packages is approximately 103.94 hours.
Complete Question:
The lifetime of a certain type of battery is normally distributed with mean value 11 hours and standard deviation 1 hour. There are nine batteries in a package. What lifetime value is such that the total lifetime of all batteries in a package exceeds that value for only 5% of all packages? (Round your answer to two decimal places.)
______ hours
Write a linear equation in standard form for the following scenario Matt is in charge of selling roses and chocolate hearts for the Valentine's Day dance he sell Beats Rose for $5 I need to Chocolate hard for $2.50 at the end of the dance he made a total of $250*
Final answer:
To write the linear equation, let x represent the number of Beats Roses sold and y represent the number of Chocolate Hearts sold. The equation is 5x + 2.50y = 250.
Explanation:
To write a linear equation in standard form for this scenario, we need to define our variables. Let x represent the number of Beats Roses sold and y represent the number of Chocolate Hearts sold.
Based on the given information, we know that the price of Beats Roses is $5, and the price of Chocolate Hearts is $2.50.
The total amount of money Matt made at the end of the dance is $250. Using this information, we can write the equation:
5x + 2.50y = 250
Therefore, the linear equation in standard form for this scenario is 5x + 2.50y = 250.
X and y are supplementary angles. Y measures 88 what is the measure of X
Answer:
x = 92
Step-by-step explanation:
Supplementary angles add to 180 degrees,
x+y = 180
We know y =88
x+88 = 180
Subtract 88 from each side
x+88-88=180-88
x =92
Find the area of the shaded sector. Round to the nearest tenth and use 3.142 or the pi button on your calculator for pi.
answers: (all answers are squared)
28.3 yd2
4.7 yd2
9.4 yd2
7.1 yd2
Answer:
The answer is option 1.
Step-by-step explanation:
It is given that the area of sector is, A = θ/360 × π × r² where r is the radius of circle. Using the formula, you are able to find the shaded sector :
θ = 90°
r = 6 yd
A = 90/360 × π × 6²
= 1/4 × π × 36
= 9π
= 28.3 yd² (near. tenth)
Answer:
The answer is 7.1yd
Step-by-step explanation:
got the wrong answer so I hope this helps yall and god bless️
There are 327 official ports of entry in the United States. The Department of Homeland Security selects 15 ports of entry at random to be audited for compliance with screening procedures of incoming travelers through the primary and secondary vehicle and pedestrian lanes. What kind of sample is this (simple random, systematic, stratified, cluster)?
Answer:
RANDOM
Step-by-step explanation:
Suppose that an accounting firm does a study to determine the time needed to complete one person's tax forms. It randomly surveys 175 people. The sample mean is 22.8 hours. There is a known population standard deviation of 6.4 hours. The population distribution is assumed to be normal. NOTE: If you are using a Student's t-distribution, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)
Construct a 90% confidence interval for the population mean time to complete the tax forms.
i. State the confidence interval. (Round your answers to two decimal places.)
ii. Calculate the error bound. (Round your answer to two decimal places.
Answer:
Step-by-step explanation:
Considering the central limit theorem, the distribution is normal since the number of samples is large. Also, the population standard
deviation is known. We would determine the z score.
To determine the z score, we subtract the confidence level from 100% to get α
α = 1 - 0.90 = 0.1
α/2 = 0.1/2 = 0.05
This is the area in each tail. Since we want the area in the middle, it becomes
1 - 0.05 = 0.95
The z score corresponding to the area on the z table is 2.05. Thus, confidence level of 90% is 1.645
Confidence interval is written in the form,
(Sample mean - margin of error, sample mean + margin of error)
The sample mean, x is the point estimate for the population mean.
Confidence interval = mean ± z × σ/√n
Where
σ = population standard Deviation
Confidence interval = x ± z × σ/√n
x = 22.8 hours
σ = 6.4 hours
n = 175
i) Confidence interval = 22.8 ± 1.645 × 6.4/√175
= 22.8 ± 0.80
The lower end of the confidence interval is
22.8 - 0.80 = 22
The upper end of the confidence interval is
22.8 + 0.80 = 23.6
ii) error bound is the same as the margin of error
Error bound = 0.8
A hand consists of 1 card drawn from a standard 52-card deck with flowers on the back and 1 card drawn from a standard 52-card deck with birds on the back. A standard deck has 13 cards from each of 4 suits (clubs, diamonds, hearts, spades). The 13 cards have face value 2 through 10, jack, queen, king, or ace. Each face value is a "kind" of card. The jack, queen, and king are "face cards."
A.How many different hands are possible? (Note that a flower-ace-of-spades, bird-queen-of-hearts and a flower-queen-of-hearts, bird-ace-of-spades are two different outcomes.)
B.How many hands consist of a pair of aces?
C.How many hands contain all face cards?
D.How many hands contain exactly 1 king?
E.How many hands consist of two of a kind (2 aces, 2 jacks, and so on)?
F.. How many hands contain at least 1 king?
Answer:
A) 2704 hands
B) 16 hands
C) 144 hands
D) 384 hands
E) 208 hands
F) 400 hands
Step-by-step explanation:
See the attached file for explanation
There are 2704 possible hands, 16 hands consist of a pair of aces, 144 hands have all face cards, 384 hands have 1 king, 208 hands consist of two of a kind, and 384 hands contain at least 1 king.
Explanation:To answer these probability related questions, we must look at the combinations which we can draw. A standard deck contains 52 cards and the hand in question consists of 1 card from 2 different decks, so:
A. The total number of different hands possible is 52 * 52 = 2704.
B. There are 4 aces in each deck so there are 4 * 4 = 16 hands that consist of a pair of aces.
C. A deck has 12 face cards and since 2 cards are being drawn, the number of hands with all face cards is 12 * 12 = 144.
D. The number of hands with exactly 1 king is found by multiplying the number of kings in a deck (4) by the number of non-kings in a deck (52-4). So, 4 * 48 = 192, but we must consider this happening in both decks: so 2 * 192 = 384.
E. For two of a kind hands (2 aces, 2 kings etc.), there are 13 kinds, thus 13 * 4 * 4 = 208.
F. For at least 1 king, either the first card or second card can be a king, so we use similar mathematics as used in D, which is 2 * 192 = 384.
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Some airlines have restrictions on the size of items of luggage that passengers are allowed to take with them. Suppose that one has a rule that the sum of the length, width and height of any piece of luggage must be less than or equal to 222 cm. A passenger wants to take a box of the maximum allowable volume. If the length and width are to be equal, what should the dimensions be?
Answer:
The dimensions should be length=width=height=74 cm.
Step-by-step explanation:
We have an optimization with restriction problem.
We have to maximize the volume, subject to restriction in the sum of the length, width and height.
Let x be the length and width, that are equal, and z be the height.
The restriction can be expressed as:
[tex]x+x+z\leq222\\\\2x+z\leq222[/tex]
We can express z in function of x as:
[tex]2x+z=222\\\\z=222-2x[/tex]
The volume, the function to be optimized, can be expressed as:
[tex]V=x^2z=x^2(222-2x)=222x^2-2x^3[/tex]
To optimize, we derive and equal to zero.
[tex]\dfrac{dV}{dx}=\dfrac{d}{dx}[222x^2-2x^3]=2*222x-3*2x^2=444x-6x^2=0\\\\\\444x-6x^2=0\\\\x(444-6x)=0\\\\444-6x=0\\\\x=444/6=74[/tex]
We have the optimum length. We can now calculate the height z:
[tex]z=222-2(74)=222-148=74[/tex]