Question

Thoroughbred Bus Company finds that its monthly costs for one particular year were given by C(t) = 10,000 t2 dollars after t months. After t months the company had P(t) = 1,000 t2 passengers per month. How fast is its cost per passenger changing after 8 months? HINT [See Example 8(b).] (Round your answer to two decimal places.) $_____________ per month

Answer 1

Explanation:

To determine the rate of change in the cost per passenger after 8 months, we must compute the derivative of the cost per passenger function, denoted as C(t)/P(t).

Firstly, ascertain the cost function C(t) by substituting the given formula: C(t) = 10,000t^2.

Following that, obtain the passenger function P(t) by inserting the provided formula: P(t) = 1,000t^2.

Next, calculate the derivative of the cost function C(t) and the passenger function P(t) concerning time, denoted as t.

C'(t) = (dC(t)/dt) = (d/dt)(10,000t^2) = 20,000t.

P'(t) = (dP(t)/dt) = (d/dt)(1,000t^2) = 2,000t.

To find the rate of change in the cost per passenger after 8 months (t = 8), substitute t = 8 into the derivative C'(t)/P'(t).

Evaluate C'(8)/P'(8) = 20,000(8) / 2,000(8) = 160,000 / 16,000 = 10.

Consequently, the cost per passenger is altering at a rate of $10 per month after 8 months.