Question

On a certain day, the total number of pieces of candy produced by a factory since it opened is modeled by C, a differentiable function of the number of hours since the factory opened. Which of the following is the best interpretation of C'(3) = 500?

a. The factory produces 500 pieces of candy during its 3rd hour of operation.
b. The factory produces 500 pieces of candy in the first 3 hours after it opens.
c. The factory is producing candy at a rate of 500 pieces per hour, 3 hours after it opens.
d. The rate at which the factory is producing candy is increasing at a rate of 500 pieces per hour per hour, 3 hours after it opens.
e. None of the above.

Answer 1

The correct interpretation of C. C'(3) = 500 is that the factory is producing candy at an instantaneous rate of 500 pieces per hour, three hours after it starts operating.

Step-by-step explanation:

The best interpretation of C'(3) = 500 is that the factory is producing candy at a rate of 500 pieces per hour, 3 hours after it opens. This is because C' represents the derivative of the candy production function with respect to time, which tells us the instantaneous rate of change of candy production. Therefore, answer choice (a), stating the factory produces 500 pieces during the third hour, is incorrect because it does not specifically relate to an instantaneous rate.

Likewise, answer choice (b) is incorrect because it implies a cumulative total over three hours instead of a rate. Option (d) is also incorrect because it implies acceleration in production rate per hour, which does not match the provided information of an instantaneous rate at a given time. The correct answer is thus (c).

Answer 2

C'(3) = 500 indicates that the factory is producing candy at a rate of 500 pieces per hour, 3 hours after it opens.

Step-by-step explanation:

The student's question concerns interpreting the derivative of a function, specifically in the context of a real-world application related to a factory's production.

The correct interpretation of C'(3) = 500 is that the factory is producing candy at a rate of 500 pieces per hour, 3 hours after it opens.

This information tells us the instantaneous rate of change of the candy production after the factory has been open for 3 hours, indicating the speed of the production line at that specific point in time.