ap cal ab unit 5 frq

Question

A graphing calculator may be used for the following problem. On a given workday, the rate, in tons per hour, at which unprocessed ore arrives at a refinery is modeled by O(t) = 20 + 10 cos ( t^2/25), where t is measured in hours and 0 ≤ t ≤ 10.

At the beginning of the workday, t = 0, the refinery has 100 tons of unprocessed ore. During the hours of operation, the refinery processes ore at a constant rate of 25 tons per hour.

a) Find O'(4). Using correct units, interpret your answer in the context of the problem.

b) How much total ore has been processed by the refinery after the plant has been open for 3 hours?

c) Is the amount of unprocessed ore at the refinery increasing or decreasing at time t = 4 hours? Show the work that leads to your answer.

d) Determine the time at which the total rate of ore at the refinery is neither increasing nor decreasing? Justify your answer.

Answer 1

a) O'(t) = -0.8*t*sin(t²/25)

O'(4) = -0.8*4*sin(16/25) = -16/5*sin(16/25) = -1.91 tons/hour².

O'(4) represents the rate of change in time (unit is tons/hour²) of the unprocessed ore arrival rate (the unit is tons/hour) at the time t = 4 hours.

b) 25 tons/hour*3 hours = 75 tons

c) Based on the result in a) the amount of unprocessed ore is decreasing at the time t = 4 hours, because O'(4) is negative.

d) The time can be found by solving O'(t) = 0 = -0.8*t*sin(t²/25). Solutions are t₁ = 0, t₂ = √(25*k*π)

Inside 0 ≤ t ≤ 10 there are 2 solutions: t = 0 and t = 5√π = 8.86 hours