Question 1 (Essay Worth 30 points)
(08.01 HC)

A yo-yo is moving up and down a string so that its velocity at time t is given by v(t) = 3cos(t) for time t ≥ 0. The initial position of the yo-yo at time t = 0 is x = 3.

Part A: Find the average value of v(t) on the interval open bracket 0 comma pi over 2 close bracket. (10 points)

Part B: What is the displacement of the yo-yo from time t = 0 to time t = π? (10 points)

Part C: Find the total distance the yo-yo travels from time t = 0 to time t = π. (10 points)

Answers

Answer 1

Part A: The average value of v(t) on the interval [0, π/2] is 0.

Part B: The displacement of the yo-yo from time t = 0 to time t = π is 0.

Part C: The total distance the yo-yo travels from time t = 0 to time t = π is 6.

Here are the calculations:

Part A: The average value of a function f(t) on the interval [a, b] is given by the following formula:

[tex]\frac{\int_a^b f(t) \, dt}{b - a}[/tex]

In this case, we have:

[tex]\frac{\int_0^{\pi/2} 3\cos(t) \, dt}{\pi/2 - 0} = \frac{3\sin(\pi/2) - 3\sin(0)}{\pi/2} = 0[/tex]

Part B: The displacement of a particle is given by the following integral:

[tex]x(t) = \int_0^t v(t) \, dt[/tex]

In this case, we have:

[tex]x(\pi) = \int_0^{\pi} 3\cos(t) \, dt = 3\sin(\pi) - 3\sin(0) = 0[/tex]

Part C: The total distance traveled by a particle is given by the following formula:

d = [tex]\int_0^t |v(t)| \, dt[/tex]

In this case, we have:

d = [tex]\int_0^{\pi} |3\cos(t)| , dt = \int_0^{\pi/2} 3\cos(t) , dt + \int_{\pi/2}^{\pi} 3\cos(t) , dt = 3\sin(\pi/2) + 3\sin(0) = 6[/tex]

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Related Questions

What is the slope of the line that is perpendicular to y = - 3x - 1

Answers

Answer:

The slope of the line perpendicular to y = -3x - 1 is 1/3.

Explanation:

To find the slope of the line perpendicular to this line, we take the negative reciprocal of -3.

The negative reciprocal of -3 is 1/3.

A school administrator believes that, in his school, the GPA of
seniors in their second semester is less than the GPA of seniors in
their first semester. The class size is large, so a census is not
practical because of time constraints. To test his belief, he selects a
SRS of senior GPA's from the first semester (n1 - 50) and an
independent SRS of senior GPA's from the second semester (n2-
50). He finds that x1 bar-3.1 and s1-0.8, and x2 bar.-2.8 and $2
-0.5. He defines p1 as the average GPA of seniors in semester 1
and u2 as the average GPA of seniors in semester 2.
The test of his belief for
statistical significance will be
a:
A. two-tailed test.
B. one-tailed left test.
C. one-tailed right test.

Answers

The test of his belief for statistical significance will be a: one-tailed right test. The correct option is C.

We can conduct a hypothesis test to verify the administrator's assertion that seniors' second-semester GPAs are lower than seniors' first-semester GPAs.

According to the null hypothesis (H0), there is no discernible difference between the two means.

H0: u1 - u2 = 0

The contrary hypothesis (Ha) would assert that the second semester's GPA is, in fact, lower than the first semester's GPA:

Ha: u1 - u2 > 0

We are using a one-tailed right test because the administrator thinks the second semester's GPA is lower.

We are eager in finding out if there is evidence to support the alternative hypothesis that the second semester's GPA is lower.

Therefore, C. One-tailed Right Test is the appropriate response.

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A credit card has an APR of 16. 42% all of last year and compounded interest daily. What was the credit cards effective interest rate last year?

Answers

The credit card last year is approximately 17.27%.

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