Answer: the square root of 72 which is approximately 8.5
Step-by-step explanation:
When draw a line diagonally in a square you get 2 right triangles so you can do pathogream theorem on them so a squared plus b squared equals c squared. In this case 6^2 + 6^2 = c^2. By solving you get the answer
The length of the pipeline running diagonally across the square field is 8.485 kilometers.
What is Pythagoras theorem?The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the sides of the square field are each 6 kilometers long. Let's sides of the square as a, b, and c, where c is the length of the diagonal (the pipeline) and a and b are the sides of the square.
Using the Pythagorean theorem, we have:
c² = a² + b²
Since the square field is a square, all sides are equal, so a = b = 6 kilometers.
c² = 6² + 6²
c² = 36+ 36
c² = 72
c = 8.485 Km
Therefore, the length of the pipeline running diagonally across the square field is 8.485 kilometers.
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how do i solve 27=(1/3)^x/2
Answer: x = -6
Step-by-step explanation: answer on edge
use the unit circle to find the value of sin(3pi/2) and cos(3pi/2). hurry I need help ASAP!
Answer:
Step-by-step explanation:
Final answer:
The value of sin(3π/2) is -1, and the value of cos(3π/2) is 0, as determined by their positions on the unit circle at an angle of 3π/2 radians.
Explanation:
To find the values of sin(3π/2) and cos(3π/2) using the unit circle, we need to understand the positions on the circle.
At 3π/2 radians, which is the same as 270°, the point on the unit circle is directly below the center of the circle, at coordinates (0, -1).
This position corresponds to a cosine value of 0 (since the x-coordinate is 0) and a sine value of -1 (since the y-coordinate is -1).
You go to the doctor and he gives you 11 milligrams of radioactive dye. After 20 minutes, 4.25 milligrams of dye remain in your system. To leave the doctor's office, you must pass through a radiation detector without sounding the alarm. If the detector will sound the alarm if more than 2 milligrams of the dye are in your system, how long will your visit to the doctor take, assuming you were given the dye as soon as you arrived? Give your answer to the nearest minute.
Rounding to the nearest minute, your visit to the doctor will take approximately 36 minutes.
To determine how long your visit to the doctor will take, we need to find the time it takes for the remaining amount of radioactive dye in your system to be less than 2 milligrams.
We can use the exponential decay formula to model the amount of radioactive dye remaining in your system after a given time:
[tex]\[ A(t) = A_0 \times e^{-kt} \][/tex]
Where:
- [tex]\( A(t) \)[/tex] is the amount of radioactive dye remaining at time ( t )
- [tex]\( A_0 \)[/tex] is the initial amount of radioactive dye
- [tex]\( k \)[/tex] is the decay constant
- [tex]\( t \)[/tex] is the time (in minutes, in this case)
Given:
- [tex]\( A_0 = 11 \)[/tex] milligrams
- [tex]\( A(t) = 4.25 \)[/tex] milligrams
-[tex]\( t = 20 \)[/tex] minutes
We need to solve for ( k ) using the given data:
[tex]\[ 4.25 = 11 \times e^{-20k} \][/tex]
First, divide both sides by 11:
[tex]\[ \frac{4.25}{11} = e^{-20k} \][/tex]
Now, take the natural logarithm of both sides:
[tex]\[ \ln\left(\frac{4.25}{11}\right) = -20k \][/tex]
Now, solve for [tex]\( k \):[/tex]
[tex]\[ k = \frac{\ln\left(\frac{4.25}{11}\right)}{-20} \][/tex]
Now that we have the decay constant, we can use it to find the time it takes for the remaining amount of dye to be less than 2 milligrams:
[tex]\[ 2 = 11 \times e^{-kt} \][/tex]
Plug in the value of ( k ) and solve for ( t ):
[tex]\[ t = \frac{\ln\left(\frac{2}{11}\right)}{-k} \][/tex]
Let's calculate ( t ):
First, let's calculate ( k ):
[tex]\[ k = \frac{\ln\left(\frac{4.25}{11}\right)}{-20} \][/tex]
[tex]\[ k \approx \frac{\ln(0.3864)}{-20} \][/tex]
[tex]\[ k \approx \frac{-0.9515}{-20} \][/tex]
[tex]\[ k \approx 0.0476 \][/tex]
Now, let's use ( k ) to find ( t ):
[tex]\[ t = \frac{\ln\left(\frac{2}{11}\right)}{-0.0476} \][/tex]
[tex]\[ t \approx \frac{\ln(0.1818)}{-0.0476} \][/tex]
[tex]\[ t \approx \frac{-1.707}{-0.0476} \][/tex]
[tex]\[ t \approx 35.81 \][/tex]
Rounding to the nearest minute, your visit to the doctor will take approximately 36 minutes.
What is the solution to X^2–64=0
Answer:
x = 8, x = -8
Step-by-step explanation:
Hello!
We can use the Difference of Squares Formula: [tex](a + b)(a - b) = a^2 - b^2[/tex]
We need the two terms to be perfect squares for this formula to work, and subtraction has to be the operation. As you can see x² is a square of x, and 64 is the square of 8.
Solve:
x² - 64 = 0(x + 8)(x - 8) = 0Set each factor to zero. (Zero Product Property)
x + 8 = 0x = -8And
x - 8 = 0x = 8The solutions are x = 8, and x = -8.
The wait times in line at a grocery store are roughly distributed normally with an average wait time of 7.6 minutes and a standard deviation of 30 seconds. What is the probability that the wait time is between 6 and 7.3 minutes
Answer:
the probability that the wait time is between 6 and 7.3 minutes = 0.725
Step-by-step explanation:
Given -
Average wait time [tex](\nu)[/tex] = 7.6 minutes
Standard deviation [tex](\sigma)[/tex] = 30 second = .5 minute
Let X be the wait times in line at a grocery store
the probability that the wait time is between 6 and 7.3 minutes
[ Put z = [tex]\frac{X - \nu }{\sigma}[/tex] ]
[tex]P (7.3> X>6 )[/tex] = [tex]P (\frac{7.3 - 7.6}{.5}> Z>\frac{6 - 7.6 }{.5} )[/tex]
= [tex](.6> Z> -3.2 )[/tex]
[Using z table]
= Area to the left of z = .6 - area to the left of z = -.32
= .7257 - .0007 = 0.725
To find the probability of a wait time between 6 and 7.3 minutes at the grocery store, convert the times to z-scores and use the standard normal distribution to find the probability between these scores.
To calculate the probability that the wait time is between 6 and 7.3 minutes for a normally distributed data set with a mean of 7.6 minutes and a standard deviation of 30 seconds, we first need to convert the wait times into z-scores. Using the formula z = (X - \\(mu\\)) / \\(sigma\\), where X is the value in the data set, \\(mu\\) is the mean, and \\(sigma\\) is the standard deviation, we find the z-scores for 6 minutes and 7.3 minutes.
The z-score for 6 minutes is z1 = (6 - 7.6) / 0.5 = -3.2, and the z-score for 7.3 minutes is z₂ = (7.3 - 7.6) / 0.5 = -0.6. Using a standard normal distribution table or calculator, we find the probabilities corresponding to these z-scores, P(z₁) and P(z₂), and subtract them to get the probability of a wait time between 6 and 7.3 minutes, which is P(z₂) - P(z₁).
a plant is already 43 centimeters tall, it will grow one centimeter every month. Let H be the plant's height (in centimeters) after M months. Then use this equation to find the plant's height after 26 months
Minimizing Packaging Costs If an open box has a square base and a volume of 107 in.3 and is constructed from a tin sheet, find the dimensions of the box, assuming a minimum amount of material is used in its construction. (Round your answers to two decimal places.)
Answer:
The dimensions of the box that minimizes the amount of material of construction is
Square base = (5.98 × 5.98) in²
Height of the box = 2.99 in.
Step-by-step explanation:
Let the length, breadth and height of the box be x, z and y respectively.
Volume of the box = xyz = 107 in³
The box has a square base and an open top.
x = z
V = x²y = 107 in³
The task is to minimize the amount of material used in its construction, that is, minimize the surface area of the box.
Surface area of the box (open at the top) = xz + 2xy + 2yz
But x = z
S = x² + 2xy + 2xy = x² + 4xy
We're to minimize this function subject to the constraint that
x²y = 107
The constraint can be rewritten as
x²y - 107 = constraint
Using Lagrange multiplier, we then write the equation in Lagrange form
Lagrange function = Function - λ(constraint)
where λ = Lagrange factor, which can be a function of x and y
L(x,y,z) = x² + 4xy - λ(x²y - 107)
We then take the partial derivatives of the Lagrange function with respect to x, y and λ. Because these are turning points, at the turning points each of the partial derivatives is equal to 0.
(∂L/∂x) = 2x + 4y - 2λxy = 0
λ = (2x + 4y)/2xy = (1/y) + (2/x)
(∂L/∂y) = 4x - λx² = 0
λ = (4x)/x² = (4/x)
(∂L/∂λ) = x²y - 107 = 0
We can then equate the values of λ from the first 2 partial derivatives and solve for the values of x and y
(1/y) + (2/x) = (4/x)
(1/y) = (2/x)
x = 2y
Hence, at the point where the box has minimal area,
x = 2y
Putting these into the constraint equation or the solution of the third partial derivative,
x²y - 107 = 0
(2y)²y = 107
4y³ = 107
y³ = (107/4) = 26.75
y = ∛(26.75) = 2.99 in.
x = 2y = 2 × 2.99 = 5.98 in.
Hence, the dimensions of the box that minimizes the amount of material of construction is
Square base = (5.98 × 5.98) in²
Height of the box = 2.99 in.
Hope this Helps!!!
To minimize packaging costs for a box with a square base and a fixed volume, we differentiate the surface area function with respect to the side length of the base and find the optimal dimensions, resulting in an approximately 5.15 inches base and a height of approximately 4.03 inches.
To minimize the material used for the open box with a square base and a given volume, we must use calculus to find the dimensions that will give us a box with minimum surface area.
Let the side of the square base be x inches and the height be h inches. Since the volume of the box is fixed at [tex]107 in^3[/tex], we have [tex]V = x^2h[/tex], which implies [tex]h = V / x^2 = 107 / x^2[/tex].
To minimize the surface area, we need to minimize the function [tex]S(x) = x^2 + 4xh[/tex]. Substituting in the equation for h, we get [tex]S(x) = x^2 + 4x(107) / x^2[/tex]. To find the minimum, take the derivative of S with respect to x and set it equal to zero. Solve for x to find the optimal dimension for the base.
After solving, we find that the dimension for the sides of the square base, x, is approximately x = 5.15 in (rounded to two decimal places). The height h would then be [tex]h = 107 / x^2[/tex] ≈ 4.03 in (rounded to two decimal places).
Mismatch went to a store spent half of her money and then $10 more she went to the second store spend half the money she has left and then $10 more then she had no money left how much money did mismatch have when she started our
Answer: I had a stroke trying to read and understand this sorry
Step-by-step explanation:
Answer:
$60.
Step-by-step explanation:
x / 2 - 10 = 0
x / 2 = 10
x = 20
Before she spent her money:
x / 2 - 10 = 20
x / 2 = 30
x = 60
She had $60 before she went shopping.
Feel free to let me know if you need more help. :)
the hypotnuse of a 45-45-90 triangle has a length of 10 units whats the length of one of its legs
Answer:
5√2 units ≈ 7.07 units
Step-by-step explanation:
The ratio of leg to hypotenuse in an isosceles right triangle is ...
leg/hypotenuse = 1/√2
Multiplying by the length of the hypotenuse, we have ...
leg = hypotenuse/√2 = 10/√2
leg = 5√2 . . . . rationalize the denominator
The length of one leg is 5√2 units, about 7.07107 units.
Final answer:
In a 45-45-90 triangle with a hypotenuse of 10 units, each leg would be of length 5√2 units.
Explanation:
The question is asking for the length of one of the legs of a 45-45-90 triangle with a hypotenuse length of 10 units. In a 45-45-90 triangle, also known as an isosceles right triangle, the lengths of the legs are equal, and each leg is √2 times smaller than the hypotenuse. To find the length of the leg (let's call it L), we set up the equation L = √2 / 2 × hypotenuse. Plugging in the hypotenuse length we get L = √2 / 2 × 10, which simplifies to L = 5√2. Thus, the length of each leg of the triangle is 5√2 units.
Which one of the following is NOT a characteristic of the sampling distribution of a sample mean? The distribution of the values is obtained by repeated sampling. The samples are all of size n. The samples are all drawn from the same population. The mean is zero and the standard deviation is one.
Answer:
The mean is zero and the standard deviation is one.
Correct, since the sample mean is not always defined by a distribution with mean 0 and deviation 1. Since the normal standard distribution is not always representative of the sample mean dsitribution
Step-by-step explanation:
For this case we want to analyze the distribution of the sample mean given by:
[tex] \bar X = \frac{\sum_{i=1}^n X_i}{n}[/tex]
Let's analyze one by one the possible options:
The distribution of the values is obtained by repeated sampling.
False, for this case we assume that the data of each observation [tex] X_i[/tex] , [tex] i =1,2,...,n[/tex] is obtained from a repated sampling method
The samples are all of size n.
False, for this case we assume that the sample mean is obtained from n data values
The samples are all drawn from the same population.
False, for this case we assume that the data comes from the same distribution
The mean is zero and the standard deviation is one.
Correct, since the sample mean is not always defined by a distribution with mean 0 and deviation 1. Since the normal standard distribution is not always representative of the sample mean dsitribution
The mean is zero and the standard deviation is one.
It is not characteristic of the sampling distribution of a sample mean.
In the Sampling Distribution of the Sample Mean , If repeated random samples of a given size n are taken from a population .where the population mean is μ and the population standard deviation is σ .then the mean of all sample means is population mean μ .
Characteristic of Sampling Distribution ,
The overall shape of the distribution is symmetric and approximately normal. There are no outliers or other important deviations from the overall pattern. The center of the distribution is very close to the true population meanSo, The mean is zero and the standard deviation is one in sampling distribution is not any mandatory condition.
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Suppose that the monthly cost, in dollars, of producing x chairs is C(x)equals0.001 x cubed plus 0.07 x squared plus 16 x plus 700, and currently 35 chairs are produced monthly. a) What is the current monthly cost?
Answer:
The current monthly cost is 1$,388.625
Step-by-step explanation:
We are given the following in the question:
[tex]C(x) = 0.001x^3 + 0.07x^2 + 16x + 700[/tex]
where C(x) is the monthly cost in dollars and x are the number of chairs produced.
Monthly chair production = 35
We have to evaluate monthly cost.
Putting x= 35 in the equation, we get,
[tex]C(35) = 0.001(35)^3 + 0.07(35)^2 + 16(35) + 700\\C(35) =1388.625[/tex]
Thus, the current monthly cost is 1$,388.625
You are permitted to use your calculator to solve an equation, find the derivative of a function at a point, or calculate the value of a definite integral. However, you must clearly indicate the setup of your question, namely the equation, function, or integral you are using. If you use other built-in features or programs, you must show the mathematical steps necessary to produce your results. Your work must be expressed in standard mathematical notation rather than calculator syntax. Show all of your work, even though the question may not explicitly remind you to do so. Clearly label any functions, graphs, tables, or other objects that you use. Justifications require that you give mathematical reasons, and that you verify the needed conditions under which relevant theorems, properties, definitions, or tests are applied. Your work will be scored on the correctness and completeness of your methods as well as your answers. Answers without supporting work will usually not receive credit. Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If your answer is given as a decimal approximation, it should be correct to three places after the decimal point. Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number.
Answer:
a. 20
b. -5
c. 5
Step-by-step explanation:
Please see attachment
An agency that hires out clerical workers claims its workers can type, on average, at least 60 words per minute (wpmwpm). To test the claim, a random sample of 50 workers from the agency were given a typing test, and the average typing speed was 58.8 wpmwpm. A one-sample tt-test was conducted to investigate whether there is evidence that the mean typing speed of workers from the agency is less than 60 wpmwpm. The resulting pp-value was 0.267.
Which of the following is a correct interpretation of the pp-value?
The probability is 0.267 that the mean typing speed is 60 wpmwpm or more for workers from the agency.
A
The probability is 0.267 that the mean typing speed is 60 wpmwpm or less for workers from the agency.
B
The probability is 0.267 that the mean typing speed is 58.8 wpmwpm or less for workers from the agency.
C
If the mean typing speed of workers from the agency is 60 wpmwpm, the probability of selecting a sample of 50 workers with mean 58.8 wpmwpm or less is 0.267.
D
If the mean typing speed of workers from the agency is less than 60 wpmwpm, the probability of selecting a sample of 50 workers with mean 58.8 wpmwpm or less is 0.267.
E
Answer:
The right answer is:
If the mean typing speed of workers from the agency is 60 wpmwpm, the probability of selecting a sample of 50 workers with mean 58.8 wpmwpm or less is 0.267.
Step-by-step explanation:
The P-value gives us the probability of getting the sample we are evaluating (in this case a sample with size n=50 and mean=58.8 wpm), if the null hypothesis is true (in this case, μ=60 wpm).
If the P-value is low enough, that is under the significance level, then we can infer that the mean that the null hypothesis states is not the actual mean, and we have evidence to reject the null hypothesis.
A teacher wants to see if a new unit on fractions is helping students learn. She has five randomly selected students take a pre-test and a post test on the material. The scores are out of 20. Suppose that you are about to compute a confidence interval for \mu_dμ d, how do you check for normality?
Answer:
Find the difference between the two scores for a number of sample distributions. Make a plot of the differences and check for outliers.
Step-by-step explanation:
Checking for Normality means basically checking if one's data distribution approximates a normal distribution.
A normal distribution is represented by a bell-shaped curve, peaking around the mean, indicating that all of the data spreads out from the mean.
Th original aim of the teacher is to check the effects of the particular added unit on the performance of students in the subject.
The teacher goes about this by testing the students before and after learning the unit.
The best way to compare of course, is to take a difference of the test scores for different samples. This first gives the idea of whether the newly introduced unit affects performance.
This set of differences is then checked for normality.
So, the best manner to make a plot of these differences. Like we mentioned earlier, a normal distribution is bell shaped. So, the plot of these differences would be a bell shaped curve if the distribution was normal and we wouldn't get a bell shaped curve if the distribution wasn't normal.
Checking for outliers help to eliminate part of data that can totally scatter the regular behaviour of the data distribution.
So, the best way for the teacher to check for normality is to find the difference between the two scores for a number of sample distributions. Make a plot of the differences and check for outliers.
Hope this Helps!!!
Since the data include the pretest and post test scores of each student or subject, then, it is approached as a paired t test. Hence, checking for normality would require obtaining the difference between the two scores for each subject, then make a plot of the differences and check for outliers.
Working on a paired sample involves taking the difference between the scores. Which is called the paired difference. The plot of the paired difference is made, and outliers are checked in the plot made. As outlier values are more associated with skewed distributions.Hence, the most appropriate check for normality in this scenario is to check for outliers in the plot made from the score difference.
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We have a bag of three biased coins a, b, and c with probabilities of coming up headsof 20%, 60%, and 80%, respectively. One coin is drawn randomly from the bag (withequal likelihood of drawing each of the three coins), and then the coin is flipped threetimes to generate the outcomes X1, X2, and X3.I.) draw the bayesian network corresponding to this setup and define the necessary cpts. Ii.) calculate which coin was most likely to have been drawn from the bag if the observed flips come out heads twice and tails once. Justify your answer.
Final answer:
To find the most likely drawn coin from a bag of three biased coins after observing two heads and one tail from three flips, we create a Bayesian network and define the Conditional Probability Tables. Then, using Bayesian inference, the likelihood of the flip sequence for each coin is calculated, allowing us to determine which coin was most likely to have been drawn.
Explanation:
The task requires calculating the likelihood of which biased coin, a, b, or c, was drawn from the bag given that the observed flip outcomes are heads twice and tails once. To approach this problem, we apply Bayesian inference.
I. Bayesian Network and Conditional Probability Tables (CPTs)
Create a Bayesian network with node C representing the choice of coin and nodes X1, X2, X3 representing the flip outcomes. Calculate the CPTs considering the bias probabilities of each coin.
II. Most Likely Coin
Using Bayes' theorem:
Calculate the likelihood of the flip sequence (HH, T) for each coin.
Determine prior probabilities (1/3 for each coin).
Compute the posterior probabilities for each coin being drawn.
Identify the highest posterior probability to conclude the most likely drawn coin.
Generally, in a situation with unequal probabilities for different outcomes, the expected long-term results will align more closely with these probabilities, influencing the most likely outcomes.
State the null and alternative hypotheses for the statistical test described below. Your answer should be an expression composed of symbols: Testing to see if there is evidence that a proportion is greater than 0.3.
H0: ____ vs Ha: ______
Answer:
Null hypothesis: [tex]p \leq 0.3[/tex]
Alternative hypothesis: [tex]p>0.3[/tex]
Step-by-step explanation:
For this question we need to take in count that the the claim that they want to test is "if the proportion is greater than 0.3". Our parameter of interest for this case is [tex]p[/tex] and the estimator for this parameter is given by this statistic [tex]\hat p[/tex] obtained from the info of sa sample obtained.
The sample proportion would be given by:
[tex] \hat p = \frac{X}{n}[/tex]
Where X represent the success and n the sample size selected
The alternative hypothesis on this case would be specified by the claim and the complement would be the null hypothesis. Based on this the system of hypothesis for this case are:
Null hypothesis: [tex]p \leq 0.3[/tex]
Alternative hypothesis: [tex]p>0.3[/tex]
And in order to check the hypothesis we can use the one sample z test for a proportion with the following statistic:
[tex] z = \frac{\hat p-p}{\sqrt{\frac{p(1-p)}{n}}}[/tex]
The null hypotheses (H₀) should be less than or equal to 0.3 and the alternative hypotheses (Hₐ) should be greater than 0.3.
What are null hypotheses and alternative hypotheses?In null hypotheses, there is no relationship between the two phenomenons under the assumption or it is not associated with the group. And in alternative hypotheses, there is a relationship between the two chosen unknowns.
Testing to see if there is evidence that a proportion is greater than 0.3.
Our parameter of interest for the case is p and the estimate for this parameter is given by the statistics [tex]\rm \hat{p}[/tex] obtained from the info of sample obtained.
The sample proportion would be given by;
[tex]\rm \hat{p} = \dfrac{X}{n}[/tex]
where X be the success and n be the sample size selected.
The alternative hypothesis, in this case, would be specified by the claim and the complement would be the null hypothesis. Based on this the system of hypotheses for this case is;
Null hypotheses: p ≤0.3
Alternative hypotheses: p > 0.3
And in order to check the hypothesis, we can use the one-sample z test for a proportion with the following statistic.
[tex]\rm z = \dfrac{\hat{p} - p}{\sqrt \dfrac{p(1-p)}{n}}[/tex]
More about the null and alternative hypotheses link is given below.
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Using traditional methods, it takes 98 hours to receive a basic driving license. A new license training method using Computer Aided Instruction (CAI) has been proposed. A researcher used the technique with 270 students and observed that they had a mean of 97 hours. Assume the standard deviation is known to be 7. A level of significance of 0.1 will be used to determine if the technique performs differently than the traditional method. Is there sufficient evidence to support the claim that the technique performs differently than the traditional method?
Answer:
Step-by-step explanation:
We would set up the hypothesis test.
For the null hypothesis,
µ = 98
For the alternative hypothesis,
µ ≠ 98
This is a 2 tailed test.
Since no population standard deviation is given, the distribution is a student's t.
Since n = 270
Degrees of freedom, df = n - 1 = 270 - 1 = 269
t = (x - µ)/(s/√n)
Where
x = sample mean = 97
µ = population mean = 98
s = samples standard deviation = 7
t = (97 - 98)/(7/√270) = - 2.35
We would determine the p value using the t test calculator. It becomes
p = 0.02
Since alpha, 0.1 > than the p value, 0.02, then we would reject the null hypothesis. Therefore, At a 10% level of significance, there sufficient evidence to support the claim that the technique performs differently than the traditional method.
A spring has natural length 23 cm. Compare the work W1 done in stretching the spring from 23 cm to 33 cm with the work W2 done in stretching it from 33 to 43 cm. (Use k for the spring constant) W
Answer:
The relation between [tex]W_{1} \ and \ W_{2}[/tex] is [tex]W_{2} = 3 \ W_{1}[/tex]
Step-by-step explanation:
Natural length = 0.23 m
Spring stretches from 23 cm to 33 cm. now
Work done [tex]W_{1}[/tex] in stretching the spring
[tex]W_{1} = \int\limits^a_b {kx} \, dx[/tex]
where b = 0 & a = 0.1 m
[tex]W_{1} = k [\frac{x^{2} }{2} ][/tex]
With limits b = 0 & a = 0.1 m
Put the values of limits we get
[tex]W_{1} = k [\frac{0.1^{2} }{2} ][/tex]
[tex]W_{1} = 0.005 k[/tex] ------- (1)
Now the work done in stretching the spring from 33 cm to 43 cm.
[tex]W_{1} = \int\limits^a_b {kx} \, dx[/tex]
With limits b = 0.1 m to a = 0.2 m
[tex]W_{2} = k [\frac{x^{2} }{2} ][/tex]
With limits b = 0.1 m to a = 0.2 m
[tex]W_{2} = k [\frac{0.2^{2} - 0.1^{2} }{2} ][/tex]
[tex]W_{2} =0.015[/tex]
[tex]\frac{W_{2} }{W_{1} } = \frac{0.015}{0.005}[/tex]
[tex]\frac{W_{2} }{W_{1} } =3[/tex]
Thus
[tex]W_{2} = 3 \ W_{1}[/tex]
This is the relation between [tex]W_{1} \ and \ W_{2}[/tex].
The work done on a spring is calculated using Hooke's Law, and it depends on the change in length of the spring (Δx) and the spring constant (k). Since the change in length is the same (10 cm) when stretching the spring from 23 cm to 33 cm (W1) and from 33 cm to 43 cm (W2), the work done during both stretches, W1 and W2, are equal.
Explanation:The work done on a spring, using Hooke's Law, is calculated with the formula W = 0.5 * k * (Δx)², where k is the spring constant and Δx is the change in length of the spring.
To find the work W1 done in stretching the spring from 23 cm to 33 cm, Δx = 33-23 = 10 cm. Thus, W1 = 0.5 * k * (10)².
The work W2 done in stretching it from 33 cm to 43 cm would be calculated similarly with Δx = 43-33 = 10 cm. Thus, W2 = 0.5 * k * (10)².
As you can see, since the stretch (Δx) is same in both cases (10 cm), W1 and W2 are equal.
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A state university finds that 115 of a random sample of 200 of its first-year students say
that "being very well-off financially is an important personal goal. If they conduct the
appropriate hypothesis test, is there evidence that a majority of all first-year students
at this university think being very well-off financially is important? Have all the
conditions been met for this situation?
Answer:
There is enough evidence to support the claim that the mayority (more than 50%) of the students think that "being very well-off financially" is an important personal goal.
The conditions are met, as this is a randome sample and the number of of positive answers (np=115) and negative answers (nq=85) are both higher than 10.
Step-by-step explanation:
The conditions to test the hypothesis are met, as this is a randome sample and the number of of positive answers (np=115) and negative answers (nq=85) are higher than 10.
The claim is that the mayority (more than 50%) of the students think that "being very well-off financially" is an important personal goal.
Then, the null and alternative hypothesis are:
[tex]H_0: \pi=0.5\\\\H_a:\pi> 0.5[/tex]
The significance level is assumed to be 0.05.
The sample has a size n=200.
The sample proportion is p=0.575.
[tex]p=X/n=115/200=0.575[/tex]
The standard error of the proportion is:
\sigma_p=\sqrt{\dfrac{\pi(1-\pi)}{n}}=\sqrt{\dfrac{0.5*0.5}{200}}\\\\\\ \sigma_p=\sqrt{0.00125}=0.0354
Then, we can calculate the z-statistic as:
[tex]z=\dfrac{p-\pi+0.5/n}{\sigma_p}=\dfrac{0.575-0.5-0.5/200}{0.0354}=\dfrac{0.0725}{0.0354}=2.0506[/tex]
This test is a right-tailed test, so the P-value for this test is calculated as:
[tex]P-value=P(z>2.0506)=0.0202[/tex]
As the P-value (0.0202) is smaller than the significance level (0.05), the effect is significant.
The null hypothesis is rejected.
There is enough evidence to support the claim that the mayority (more than 50%) of the students think that "being very well-off financially" is an important personal goal.
Two candidates ran for class president. The candidate that won received 80% of the 290 total votes. How many votes did the winning candidate receive?
Answer:
232 votes
Step-by-step explanation:
This question is just asking what 80% of 290 is. In order to find this, you convert 80% into a decimal, then multiply by 290.
Converting percentages into decimals is just moving the decimal left twice.
0.8 x 290 = 232
You can check this by doing 232/290, and making sure it's 80%.
I hope this helps!
What is the value of x in the equation 6(x + 1) – 5x = 8 + 2(x - 1)?
OIT MT
Answer:
x=0
Step-by-step explanation:
To solve, we need to get all the variables on one side of the equation, and all the numbers on the other
6(x + 1) – 5x = 8 + 2(x - 1)
First, distribute the 6 on the left
6*x+6*1 -5x=8 +2(x-1)
6x+6-5x=8+2(x=1)
Combine like terms on the left
(6x-5x)+6=8+2(x-1)
x+6=8+2(x-1)
Distribute the 2 on the right
x+6=8+2*x+2*-1
x+6=8+2x-2
Combine like terms on the right
x+6=2x+(8-2)
x+6=2x+6
Subtract x from both sides
6=x+6
Subtract 6 from both sides
x=0
Hope this helps! :)
Which conversion factors can be used to multiply 4 kilometers/min to get meters per hour
Answer:
(1000 m)/(1 km) and (60 min)/(1 h)
Step-by-step explanation:
Put the unit you don't want in a position to cancel the unit in the given number. Write the fraction so that the equivalent amount of the unit you do want is on the other side of the fraction bar.
Here we have km/min with km in the numerator. To cancel that, we need a fraction with km in the denominator. We want meters (m) in the numerator, so we need a fraction that has a number of meters equivalent to 1 km. That will be ...
(1000 m)/(1 km)
This is one of the conversion factors we will need to multiply by.
__
We also have "min" in the denominator. To cancel that, we need a conversion factor with min in the numerator. The unit we want in the denominator is h (hours), so we need an equivalent for hours and minutes. That would be 60 min = 1 h, so we write the conversion factor as ...
(60 min)/(1 h)
So, our conversion factors are ...
(1000 m)/(1 km) and (60 min)/(1 h)
_____
The converted number is ...
(4 km/min)(1000 m/km)(60 min/h) = 240,000 m/h
__
Comment on conversion factors
As you can see, we write the fraction so equal amounts are in numerator and denominator. Since the amounts are equal, the value of the fraction is 1. Multiplying by 1 in this form doesn't change the original value, it only changes the units.
L'Shanda can choose between 3 sweaters and 4 skirts. If she selects 1 sweater and 1 skirt, how many possible outcomes are in the sample space?
The answer is 12 because 3 x 4 is 12
Simplify to create equivalent to 19-6(-k+4)
Answer: don’t use k for the other numbers
Step-by-step explanation:
Answer:
6k -5
Step-by-step explanation:
19-6(-k+4)
Distribute
19 -6*-k -6*4
19 +6k -24
Combine like terms
6k +19-24
6k -5
Which decimal is terminating? 0.12 repeating 0.4444... 0.56 repeating 0.7878
0.7878 is terminating.
What is terminating decimal?Terminating decimals are the numbers that have a fixed or a finite number of digits after the decimal point. Decimal numbers are used to represent the partial amount of whole, just like fractions. In this lesson, we will focus on the type of decimal numbers, that is, terminating decimal numbers.
Terminating decimal has finite digits and non-terminating decimals do not have finite digits.It is easy to represent a terminating decimal in the form of p/q but it is difficult to express a non-terminating decimal (non-repeating) in p/q form, where q is not equal to 0.As, per the definition
0.12 repeats
0.44 repeats
0.56 repeats
But, 0.7878 does not repeat and get terminated.
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What is the surface area of the box if it is scaled up by a factor of 10?
Height 20in
W 28in
L 50in
Answer:
it’s everything multiplied by 10
so
200
280
500
Step-by-step explanation:
Suppose you are working in an insurance company as a statistician. Your manager asked you to check police records of car accidents and out of 576 accidents you selected randomly, teenagers were at the wheel in 120 of them. (a) Construct a 95% confidence interval for the percentage of all auto accidents that involve teenage drivers. (b) Explain your findings to your manager. (c) Before you were hired in the company, the percentage of teenagers who where involved in car accidents was 18%. Your manager wants you to see if the percentage of teenagers has changed since you join the company. Perform an appropriate hypothesis test using α = 0.05 and Interpret your results to your manager. (d) Use α = 0.1 and repeat part (c).
Answer:
(a) 95% confidence interval for the percentage of all car accidents that involve teenage drivers is [0.177 , 0.243].
(b) We are 95% confident that the percentage of all car accidents that involve teenage drivers will lie between 17.7% and 24.3%.
(c) We conclude that the the percentage of teenagers has not changed since you join the company.
(d) We conclude that the the percentage of teenagers has changed since you join the company.
Step-by-step explanation:
We are given that your manager asked you to check police records of car accidents and out of 576 accidents you selected randomly, teenagers were at the wheel in 120 of them.
(a) Firstly, the pivotal quantity for 95% confidence interval for the population proportion is given by;
P.Q. = [tex]\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} }}[/tex] ~ N(0,1)
where, [tex]\hat p[/tex] = sample proportion teenage drivers = [tex]\frac{120}{576}[/tex] = 0.21
n = sample of accidents = 576
p = population percentage of all car accidents
Here for constructing 95% confidence interval we have used One-sample z proportion statistics.
So, 95% confidence interval for the population population, p is ;
P(-1.96 < N(0,1) < 1.96) = 0.95 {As the critical value of z at 2.5% level
of significance are -1.96 & 1.96}
P(-1.96 < [tex]\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} }}[/tex] < 1.96) = 0.95
P( [tex]-1.96 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} }}[/tex] < [tex]{\hat p-p}[/tex] < [tex]1.96 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} }}[/tex] ) = 0.95
P( [tex]\hat p-1.96 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} }}[/tex] < p < [tex]\hat p+1.96 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} }}[/tex] ) = 0.95
95% confidence interval for p = [[tex]\hat p-1.96 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} }}[/tex] , [tex]\hat p+1.96 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} }}[/tex]]
= [ [tex]0.21-1.96 \times {\sqrt{\frac{0.21(1-0.21)}{576} }}[/tex] , [tex]0.21+1.96 \times {\sqrt{\frac{0.21(1-0.21)}{576} }}[/tex] ]
= [0.177 , 0.243]
Therefore, 95% confidence interval for the percentage of all car accidents that involve teenage drivers is [0.177 , 0.243].
(b) We are 95% confident that the percentage of all car accidents that involve teenage drivers will lie between 17.7% and 24.3%.
(c) We are also provided that before you were hired in the company, the percentage of teenagers who where involved in car accidents was 18%.
The manager wants to see if the percentage of teenagers has changed since you join the company.
Let p = percentage of teenagers who where involved in car accidents
So, Null Hypothesis, [tex]H_0[/tex] : p = 18% {means that the percentage of teenagers has not changed since you join the company}
Alternate Hypothesis, [tex]H_A[/tex] : p [tex]\neq[/tex] 18% {means that the percentage of teenagers has changed since you join the company}
The test statistics that will be used here is One-sample z proportion statistics;
T.S. = [tex]\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} }}[/tex] ~ N(0,1)
where, [tex]\hat p[/tex] = sample proportion teenage drivers = [tex]\frac{120}{576}[/tex] = 0.21
n = sample of accidents = 576
So, test statistics = [tex]\frac{0.21-0.18}{\sqrt{\frac{0.21(1-0.21)}{576} }}[/tex]
= 1.768
The value of the sample test statistics is 1.768.
Now at 0.05 significance level, the z table gives critical value of -1.96 and 1.96 for two-tailed test. Since our test statistics lies within the range of critical values of z, so we have insufficient evidence to reject our null hypothesis as it will not fall in the rejection region due to which we fail to reject our null hypothesis.
Therefore, we conclude that the the percentage of teenagers has not changed since you join the company.
(d) Now at 0.1 significance level, the z table gives critical value of -1.6449 and 1.6449 for two-tailed test. Since our test statistics does not lie within the range of critical values of z, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region due to which we reject our null hypothesis.
Therefore, we conclude that the the percentage of teenagers has changed since you join the company.
The graph of the equation below is a circle. What is the length of the radius of the circle?
(X-4)2+(y+12)2=17^2
Answer:
17
Step-by-step explanation:
(X-4)2+(y+12)2=17^2
radius is square root of 17^2
9. Poppy is using wire to make metal wall hang
radius shown for her friends. Her older sist
hangings with a diameter that is 1 inches longer
How much more wire did her sister use per
Poppy? Use 3.14 for T. Write your answer a
the nearest hundredth.
•6.25
wall hangings that have the
sister is making her wall
ches longer than Poppy's.
eper wall hanging than
nswer as a decimal rounded to the nearest hundredth.
The difference in wire required per wall hanging, rounded to the nearest hundredth, is roughly 10.99 inches.
Given that Poppy is using a metal wire wall hanging with radius of 6.25 inches, his sister is making same with radius 1 3/4 inches longer than the Poppy.
We need to find how much more wire did she need.
Calculating the circumference of each wall hanging will help us determine how much wire Poppy's sister used in comparison to Poppy.
The radius of Poppy's wall hanging is 6.25 inches, hence the following formula can be used to determine its circumference:
C = 2πr, where r is the radius.
Putting the values,
C = 2 x 3.14 x 6.25
C ≈ 39.25 inches
The diameter of Poppy's sister's wall hanging is 1 3/4 inches larger than Poppy's. Her wall hanging's radius would be 6.25 + 1 3/4 inches.
Converting 1 3/4 to an improper fraction: 1 3/4 = 7/4
Radius of Poppy's sister's wall hanging = 6.25 + 7/4
Radius ≈ 6.25 + 1.75
Radius ≈ 8 inches
Now we can calculate the circumference of Poppy's sister's wall hanging:
C = 2 x 3.14 x 8
C ≈ 50.24 inches
By deducting Poppy's circumference from Poppy's sister's circumference, we can calculate the difference in wire utilized per wall hanging:
Difference = Poppy's sister's circumference - Poppy's circumference
Difference ≈ 50.24 - 39.25
Difference ≈ 10.99 inches
Hence the difference in wire used per wall hanging is approximately 10.99 inches.
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The diagram below shows the measurements of two identical rectangular prisms joined together. Austin says the combined volume of the prisms is 225 cubic meters. Explain Austin’s error and give the correct volume.
Answer:
450 cubic meters
Step-by-step explanation:
( 15 X 5 X 3) + ( 15 x 5 x 3)225 + 225 450 cubic metersAustin didn't add the 2 prisms, he multiplied the 1st prism and kept it as the answer.
Austin is not correct as the combined volume of the prisms is 1100 cubic meter.
What is Volume?Volume is a three-dimensional quantity used to calculate a solid shape's capacity. That means that the volume of a closed form determines how much three-dimensional space it can fill.
From the figure the dimension of prism is 5 x 11 x 10.
Austin says the combined volume of the prisms is 225 cubic meters.
So, Volume of combined prism
= 2 x 5 x 11 x 10
= 2 x 55 x 10
= 110 x 10
= 1100 cubic meter.
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