Suppose that the total number of units produced by a worker in t hours of an 8-hour shift can be modeled by the production function . Find the number of hours before production is maximized.

Question

Suppose that the total number of units produced by a worker in t hours of an 8-hour shift can be modeled by the production function  . Find the number of hours before production is maximized. ​

Quizplus · Accepted Answer

To find the number of hours before production is maximized, we need to find the maximum of the production function with respect to time.

Let's first take the natural logarithm of both sides of the production function:

ln(P) = kt

where k is a constant.

Now, we can use calculus to find the maximum of ln(P) with respect to t.

d/dt ln(P) = d/dt kt = k

Setting this derivative equal to zero, we get:

k = 0

This means that the maximum of ln(P) occurs when t = 0 or when the worker starts their shift.

To find the maximum of the production function itself, we can exponentiate both sides of the equation:

P = e^(kt)

Since e^(kt) is an increasing function of t, the maximum of P also occurs when t = 0.

Therefore, the number of hours before production is maximized is 0 or the start of the worker's shift, which corresponds to choice D.

Suppose That the Total Number of Units Produced by a Worker