Answer:
Check step-by-step-explanation.
Step-by-step explanation:
A given criteria for geometric series of the form [tex]\sum_{n=0}^{\infty} r^n[/tex] is that [tex]|r|<1[/tex]. Other wise, the series diverges. When it converges, we know that
[tex] \sum_{n=0}^\infty r^n = \frac{1}{1-r}[/tex].
So,
a)[tex]\sum_{n=0}^\infty (\frac{3}{2})^n[/tex] diverges since [tex]\frac{3}{2}>1[/tex]
b)[tex]\sum_{n=0}^\infty (\frac{1}{2})^n [/tex]converges since [tex]\frac{1}{2}<1[/tex], and
[tex]\sum_{n=0}^\infty (\frac{1}{2})^n= \frac{1}{1-\frac{1}{2}} = \frac{2}{2-1} = 2[/tex]
c)We can use the series in b) but starting at n=1 instead of n=0. Since they differ only on one term, we know it also converges and
[tex]\sum_{n=1}^{\infty}(\frac{1}{2})^n = \sum_{n=0}^{\infty}(\frac{1}{2})^n-(\frac{1}{2})^0 = 2-1 = 1[/tex].
d)Based on point c, we can easily generalize that if we consider the following difference
[tex]\sum_{n=1}^\infty r^n-\sum_{n=0}^\infty r^n = r^0 = 1[/tex]
So, they differ only by 1 if the series converges.
Final answer:
A divergent geometric series has a common ratio (r) greater than 1. A convergent geometric series has a common ratio (r) between -1 and 1. The sums of geometric series that start at n = 0 and n = 1 are different because the first term is included or omitted.
Explanation:
The questions can be answered as -
(a) A geometric series that diverges is an example where the common ratio (r) is greater than 1. An example of a divergent geometric series is: 2 + 4 + 8 + 16 + ...
(b) A geometric series that converges is an example where the common ratio (r) is between -1 and 1. An example of a convergent geometric series starting at n = 0 is: 1 - 1/2 + 1/4 - 1/8 + ... To find its sum, we can use the formula for the sum of a geometric series: S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio. Plugging in the values, the sum of this series is 2/3.
(c) A geometric series that starts at n = 1 and converges can have a different sum since the first term is omitted from the calculation. An example of such a series is: 1/2 + 1/4 + 1/8 + 1/16 + ... To find its sum, we use the same formula as in part (b) but with a different first term. In this case, the sum of the series is 1/2.
(d) The sums for a geometric series that starts at n = 0 and n = 1 are different because the first term is included in the sum for n = 0 but omitted in the sum for n = 1.
Which is the exponential form shown 2/5 2/5 2/5 2/5
Answer:
[tex] \bigg(\frac{2}{5} \bigg)^{4} [/tex]
Step-by-step explanation:
[tex] \frac{2}{5} \times \frac{2}{5} \times \frac{2}{5} \times \frac{2}{5} = \bigg(\frac{2}{5} \bigg)^{4} \\ [/tex]
In the line marked *, what strategy was used?
combining like terms
fills in the orange box.
reducing a fraction
renaming a fraction
Answer:
Renaming a fraction
Step-by-step explanation:
Answer:
1. renaming a fraction
2. de-c
Step-by-step explanation:
The diameter of a circle is 9 centimeters. Find the area to the nearest tenth.
Answer:
63.6 square cm
Step-by-step explanation:
[tex]d = 9 \: cm \implies \: r = 4.5 \: cm \\ area \: of \: circle \\ = \pi {r}^{2} \\ = 3.14 \times ( {4.5})^{2} \\ = 3.14 \times 20.25 \\ = 63.585 \\ = 63.6 \: {cm}^{2} \\ [/tex]
It has long been stated that the mean temperature of humans is 98.6degreesF. However, two researchers currently involved in the subject thought that the mean temperature of humans is less than 98.6degreesF. They measured the temperatures of 44 healthy adults 1 to 4 times daily for 3 days, obtaining 200 measurements. The sample data resulted in a sample mean of 98.3degreesF and a sample standard deviation of 1 degrees F.
a. Use the P-value approach to conduct a hypothesis test to judge whether the mean temperature of humans is less than 98.6 degrees F at the α= 0.01 level of significance.
b. State the hypotheses.
Answer:
We conclude that the mean temperature of humans is more than or equal to 98.6°F.
Step-by-step explanation:
We are given that it has long been stated that the mean temperature of humans is 98.6°F. However, two researchers currently involved in the subject thought that the mean temperature of humans is less than 98.6°F.
They measured the temperatures of 44 healthy adults 1 to 4 times daily for 3 days, obtaining 200 measurements. The sample data resulted in a sample mean of 98.3°F and a sample standard deviation of 1°F.
Let [tex]\mu[/tex] = true mean temperature of humans.
SO, Null Hypothesis, [tex]H_0[/tex] : [tex]\mu \geq[/tex] 98.6°F {means that the mean temperature of humans is more than or equal to 98.6°F}
Alternate Hypothesis, [tex]H_A[/tex] : [tex]\mu[/tex] < 98.6°F {means that the mean temperature of humans is less than 98.6°F}
The test statistics that will be used here is One-sample t test statistics as we don't know about the population standard deviation;
T.S. = [tex]\frac{\bar X -\mu}{\frac{s}{\sqrt{n} } }[/tex] ~ [tex]t_n_-_1[/tex]
where, [tex]\bar X[/tex] = sample mean temperature of 44 adults = 98.3°F
s = sample standard deviation = 1°F
n = sample of healthy adults = 44
So, test statistics = [tex]\frac{98.3-98.6}{\frac{1}{\sqrt{43} } }[/tex] ~ [tex]t_4_3[/tex]
= -1.967
Hence, the value of test statistics is -1.967.
Now, P-value of the test statistics is given by;
P-value = P( [tex]t_4_3[/tex] < -1.967) = 0.029 or 2.9%
If the P-value of test statistics is more than the level of significance, then we will not reject our null hypothesis as it will not fall in the rejection region.If the P-value of test statistics is less than the level of significance, then we will reject our null hypothesis as it will fall in the rejection region.
Now, here the P-value is 0.029 which is clearly higher than the level of significance of 0.01, so we will not reject our null hypothesis as it will not fall in the rejection region.
Therefore, we conclude that the mean temperature of humans is more than or equal to 98.6°F.
For the given value, indicates whether the inequality is true or false.
4. 13 - x< 4; x = 9
5. 45 < 2x - 5; x = 20
Answer:
4. False
5. False
Step-by-step explanation:
4. Plug in 9 for x in the inequality:
13 - 9 < 4
4 < 4
We see that 4 is actually not less than 4, but is equal. So, this inequality is FALSE.
5. Plug in 20 for x in the inequality:
45 < 2 * 20 - 5
45 < 40 - 5
45 < 35
Obviously, 45 is larger than 35, so this is also FALSE.
Hope this helps!
Answer:
Both false
Step-by-step explanation:
4.
Substitute 9 in for x in the inequality
13 - x< 4
13 - 9< 4
Subtract
4< 4
4 is not less than 4, but is equal, so this is false.
5.
Substitute 20 in for x in the inequality
45 < 2x - 5
45 < 2(20) - 5
Multiply
45 < 40 - 5
Subtract
45 < 35
Since 35 is not greater than 45, this is false.
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
9514 1404 393
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.
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:
A bus arrives at a bus stop at a randomly selected time within a 1-hour period. A passenger arrives at the bus stop at a randomly selected time with the same hour. The passenger is willing to wait for the bus for up to 1/4 of an hour. What is the probability that the passenger will catch the bus?
Answer:
75%
Step-by-step explanation:
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]
A fan is marked up 40% on the original price. The original price was $20. What is the new price of the fan before sales tax?
Answer:
$28
Step-by-step explanation:
The markup was 0.40×$20 = $8, so the new price is ...
$20 +8 = $28
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
To learn more about probability refer to:
https://brainly.com/question/25870256
#SPJ2
How do u convert 3 and 4/7 to an improper fraction?
Answer:
3 4/7 = 25/7
Step-by-step explanation:
multiply the whole number(3) by the denominator(7).
Then you have 21.
Add the product of the last 2 numbers to the numerator(4)
21 + 4 = 25
Note: (denominator x whole number) + numerator
We are required to convert 3 and 4/7 to an improper fraction.
The fraction 3 and 4/7 to an improper fraction is 25 / 7
An improper fraction is a fraction in which the numerator is greater than the denominator.Numerator refers to the upper number in a fractionDenominator refers to the bottom number in a fraction.Given:
3 4/7
= {7(3) + 4} / 7
= (21 + 4) / 7
= 25 / 7
Therefore, the fraction 3 and 4/7 to an improper fraction is 25 / 7
Read more:
https://brainly.com/question/387381
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?
9514 1404 393
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.
At a research facility that designs rocket engines, researchers know that some engines fail to ignite as a result of fuel system error. From a random sample of 40 engines of one design, 14 failed to ignite as a result of fuel system error. From a random sample of 30 engines of a second design, 9 failed to ignite as a result of fuel system error. The researchers want to estimate the difference in the proportion of engine failures for the two designs. Which of the following is the most appropriate method to create the estimate?
a. A one-sample z-interval for a sample proportion
b. A one-sample z-interval for a population proportion
c. A two-sample z-interval for a population proportion
d. A two-sample z-interval for a difference in sample proportions
e. A two-sample z-interval for a difference in population proportions
Answer:
d) A two-sample z-interval for a difference in sample proportions
Step-by-step explanation:
Explanation:-
Given data a random sample of 40 engines of one design, 14 failed to ignite as a result of fuel system error.
First sample proportion [tex]p_{1} = \frac{14}{40} = 0.35[/tex]
Given data random sample of 30 engines of a second design, 9 failed to ignite as a result of fuel system error.
second sample proportion [tex]p_{2} = \frac{9}{30} = 0.30[/tex]
Null hypothesis: H₀: Assume that there is no significant between the two designs
H₀: p₁ = p₂
Alternative Hypothesis: H₁:
H₁: p₁ ≠ p₂
The test statistic
[tex]Z = \frac{p_{1}-p_{2} }{\sqrt{p q(\frac{1}{n_{1} } + \frac{1}{n_{2} } } )}[/tex]
where [tex]p = \frac{n_{1} p_{1}+n_{2} p_{2} }{n_{1} +n_{2} }[/tex]
q =1-p
Answer:
E
Step-by-step explanation:
A two-sample z-interval for a difference in population proportions
Here's a fun question: In 10 years, the total age of 2 brothers and 2 sisters will be 100 years. What will be their total age in 7 years?
PS, the answer is not 94
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)
The weights of 6-week-old poults (juvenile turkeys) are normally distributed with a mean 8.9 pounds and standard deviation 1.9 pounds. A turkey farmer wants to provide a money-back guarantee that her 6-week poults will weigh at least a certain amount. What weight should she guarantee so that she will have to give her customer's money back only 1% of the time?
A) 4.47 lb
B) 4.02 lb
C) 4.92 lb
D) 3.58 lb
Answer:
[tex]z=-2.33<\frac{a-8.9}{1.9}[/tex]
And if we solve for a we got
[tex]a=8.9 -2.33*1.9=4.47[/tex]
And the best answer for this case would be:
A) 4.47 lb
Step-by-step explanation:
Let X the random variable that represent the weights of juvenile turkeys, and for this case we know the distribution for X is given by:
[tex]X \sim N(8.9,1.9)[/tex]
Where [tex]\mu=8.9[/tex] and [tex]\sigma=1.9[/tex]
The z score formula very useful for this case is given by:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
For this part we want to find a value a, such that we satisfy this condition:
[tex]P(X>a)=0.99[/tex] (a)
[tex]P(X<a)=0.01[/tex] (b)
Both conditions are equivalent on this case. We can use the z score again in order to find the value a.
As we can see on the figure attached the z value that satisfy the condition with 0.01 of the area on the left and 0.99 of the area on the right it's z=-2.33. On this case P(Z<-2.33)=0.01 and P(z>-2.33)=0.99
If we use condition (b) from previous we have this:
[tex]P(X<a)=P(\frac{X-\mu}{\sigma}<\frac{a-\mu}{\sigma})=0.01[/tex]
[tex]P(z<\frac{a-\mu}{\sigma})=0.01[/tex]
So we have this relation
[tex]z=-2.33<\frac{a-8.9}{1.9}[/tex]
And if we solve for a we got
[tex]a=8.9 -2.33*1.9=4.47[/tex]
And the best answer for this case would be:
A) 4.47 lb
Miguel is selling tickets to a barbecue. Adult tickets cost $7.00 and children's tickets cost $5.00. He sells six more children's tickets than adult tickets. The total amount of money he collects is $234.00. How many adult tickets and how many children's tickets did he sell?
Answer:
17 adult tickets
23 children's tickets
Step-by-step explanation:
Adult tickets 17x7=119
Child tickets 23x5=115 23 is 6 more than 17.
119+115=234
Graph this function:
y=5x
Click to select points on the graph.
Answer:
You did not include your graph, therefore I will explain in the step-by-step everything you should need to know.
Step-by-step explanation:
The standard slope formula is y=mx + b, where m = slope (rise/run) and b = the y intercept.
Using this knowledge, we know that this line has a slope of 5, or 5/1, which means that you will go up 5 and 1 to the right. Since there is no + b part, that means b = 0, so the line begins at 0,0. So, using this information, possible points include 0,0, 5,1, 10,2, 15,3, etc.
The local supermarket buys lettuce each day to ensure really fresh produce. Each morning any lettuce that is left from the previous day is sold to a dealer that resells it to farmers who use it to feed their animals. This week the supermarket can buy fresh lettuce for $10.00 a box. The lettuce is sold for $21.00 a box and the dealer that sells old lettuce is willing to pay $2.00 a box. Past history says that tomorrow's demand for lettuce averages 262 boxes with a standard deviation of 43 boxes. How many boxes of lettuce should the supermarket purchase tomorrow
Given Information:
Cost price of lettuce = $10
Selling price of lettuce = $21
Salvage value = $2
Average demand of lettuce = μ = 262 boxes
Standard deviation of lettuce = σ = 43 boxes
Required Information:
How many boxes of lettuce should the supermarket purchase = ?
Answer:
n = 271 boxes
Step-by-step explanation:
The required number of lettuce boxes that supermarket should purchase is given by
n = μ + (z-score)σ
Where μ is the average demand of lettuce boxes, σ is the standard deviation, and z-score can be calculated by
p = C_us/(C_us + C_os)
Where the cost of under stocking is given by
C_us = Selling price of lettuce - Cost price of lettuce
C_us = $21 - $10
C_us = $11
The cost of over stocking is given by
C_os = Cost price of lettuce - Salvage value
C_os = $10 - $2
C_os = $8
p = C_us/(C_us + C_os)
p = 11/(11 + 8)
p ≈ 58%
The z-score corresponding to 58% is 0.202
n = 262 + (0.202)43
n = 270.68
n = 271 boxes
Therefore, the supermarket should purchase 271 boxes of lettuce tomorrow.
Anna is three times as old as Diane.If the sum of their ages is 44,how old is Anne? ( Use d as the variable )
Answer:
Anna33 diane11
Step-by-step explanation:
11×3=33
33+11=44
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
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.
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%.
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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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.
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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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
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.
Three airlines serve a small town in Ohio. Airline A has 50% of all the scheduled flights, airline B has 30%, and Airline C has 20%. Their on-time rates are 80%, 65%, and 40%, respectively. A plane has just left on-time. what is the overall probability of leaving on-time
Answer:
185
Step-by-step explanation:
cause if you add up 80+65+40 right?
To find the overall probability of leaving on-time, we consider the probabilities of each airline and their respective on-time rates. The weighted average of the on-time rates is calculated using the percentages of scheduled flights for each airline. Summing up the weighted contributions gives us the overall probability of leaving on-time.
Explanation:To find the overall probability of leaving on-time, we need to consider the probabilities of each airline and their respective on-time rates. First, we calculate the weighted average of the on-time rates using the percentages of scheduled flights for each airline.
Airline A contributes 50% of the flights with an 80% on-time rate, so its weighted contribution is 0.5 x 0.8 = 0.4.
Airline B contributes 30% of the flights with a 65% on-time rate, so its weighted contribution is 0.3 x 0.65 = 0.195.
Airline C contributes 20% of the flights with a 40% on-time rate, so its weighted contribution is 0.2 x 0.4 = 0.08.
To find the overall probability, we sum up the weighted contributions: 0.4 + 0.195 + 0.08 = 0.675.
Therefore, the overall probability of leaving on-time is 0.675 or 67.5%.
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