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A parking garage has 230 cars in it when it opens at 8???????? (???? = 0). On the interval 0 ≤ ???? ≤ 10, cars enter the parking garage at the rate ???? ′ (????) = 58 cos(0.1635???? − 0.642) cars per hour and cars leave the parking garage at the rate ???? ′ (????) = 65 sin(0.281????) + 7.1 cars per hour (a) How many cars enter the parking garage over the interval ???? = 0 to ???? = 10 hours? (b) Find ????′′(5). Using correct units, explaining the meaning of this value in context of the problem. (c) Find the number of cars in the parking garage at time ???? = 10. Show the work that leads to your answer. (d) Find the time ???? on 0 ≤ ???? ≤ 10, when the number of cars in the parking garage is a maximum. To the nearest whole number, what is the maximum number of cars in the parking garage? Justify your answer

User Allende
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2 Answers

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Final answer:

The question addresses finding the total number of cars entering a parking garage, the rate of change of cars at a specific time, the total number of cars at a certain time, and determining when the number of cars in the garage is at its maximum.

Step-by-step explanation:

The question involves calculus and concerns the rates at which cars enter and leave a parking garage, the net change in car count, and the maximum number of cars in the garage during a specific timeframe.

To answer part (a), which asks how many cars enter the parking garage over the interval from t = 0 to t = 10 hours, we calculate the integral of the entry rate function: ∫ 58 cos(0.1635t - 0.642) dt from 0 to 10. For part (b), to find v''(5), we differentiate the net rate of change of cars (which is the difference between entry and exit rates) and evaluate it at t = 5. This represents the rate of acceleration of car count at t = 5 hours. Part (c) requires us to calculate the number of cars at t = 10 by integrating the net rate of change of the number of cars (entry rate minus exit rate) from 0 to 10 and adding this to the initial 230 cars. For part (d), to determine the time t when the number of cars is maximum, we look for critical points by setting the derivative of the net rate of change to zero and solve for t in the interval [0, 10]. We then test these critical points to find which one gives the maximum number of cars.

User SheetJS
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User Murat Erdogan
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