Explaining How to Compare Water Levels Ericka decided to compare her observation to the average annual trend, which shows the wa
ter rising 1.8 mm/year. Remember, she used 6.2 years as her time period. Explain how she would calculate the difference between how much water levels rose on average and how much the water level fell in the part of the river she observed.
2 answers:
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Part A - The average value of v(t) over the interval (0, π/2) is 6/π
Part B - The displacement of the yo-yo from time t = 0 to time t = π is 0 m
Part C - The total distance the yo-yo travels from time t = 0 to time t = π is 6 m.
<h3>Part A: Find the average value of v(t) on the interval (0, π/2)</h3>The average value of a function f(t) over the interval (a,b) is
So, since velocity at time t is given by v(t) = 3cos(t) for time t ≥ 0. Its average value over the interval (0, π/2) is given by
Since v(t) = 3cost, we have
So, the average value of v(t) over the interval (0, π/2) is 6/π
<h3>Part B: What is the displacement of the yo-yo from time t = 0 to time t = π?</h3>To find the displacement of the yo-yo, we need to find its position.
So, its position x = ∫v(t)dt
= ∫3cos(t)dt
= 3∫cos(t)dt
= 3sint + C
Given that at t = 0, x = 3. so
x = 3sint + C
3 = 3sin0 + C
3 = 0 + C
C = 3
So, x(t) = 3sint + 3
So, its displacement from time t = 0 to time t = π is
Δx = x(π) - x(0)
= 3sinπ + 3 - (3sin0 + 3)
= 3 × 0 + 3 - 0 - 3
= 0 + 3 - 3
= 0 + 0
= 0 m
So, the displacement of the yo-yo from time t = 0 to time t = π is 0 m
<h3>Part C: Find the total distance the yo-yo travels from time t = 0 to time t = π. (10 points)</h3>The total distance the yo-yo travels from time t = 0 to time t = π is given by
So, the total distance the yo-yo travels from time t = 0 to time t = π is 6 m.
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