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 the rate of production is maximized. That is, find the point of diminishing returns.

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 the rate of production is maximized. That is, find the point of diminishing returns. ​​

Quizplus · Accepted Answer

To find the point of diminishing returns, we need to find where the rate of production, or the derivative of the production function, is maximized. 
Taking the derivative of the given production function, we get: 
dP/dt = 11e^(0.2t) * 0.2 
Setting this equal to 0 and solving for t, we get: 
0 = 11e^(0.2t) * 0.2 
0 = e^(0.2t) 
ln(0) = 0.2t 
This is undefined, so we cannot have a maximum rate of production when t=0. 
Taking the second derivative of the production function, we get: 
d^2P/dt^2 = 11e^(0.2t) * (0.2)^2 
Since this is always positive for any value of t, we know that the production function is concave up and there are no inflection points. Therefore, the point of diminishing returns occurs at the latest possible time, which is when the worker has worked the full 8-hour shift. So, the number of hours before the rate of production is maximized is 8-t. 
Plugging this into the given production function, we get: 
P = 11e^(0.2(8-t)) 
Simplifying this, we get: 
P = 11e^(1.6-0.2t) 
This function represents the total number of units produced when the worker has worked for 8-t hours. To find the point of diminishing returns, we need to find the value of t that maximizes this function. 
Taking the derivative of this function, we get: 
dP/dt = -2.2e^(1.6-0.2t) 
Setting this equal to 0 and solving for t, we get: 
0 = -2.2e^(1.6-0.2t) 
0 = e^(1.6-0.2t) 
ln(0) = 1.6-0.2t 
This is undefined, so we cannot have a maximum rate of production when t=8. 
Taking the second derivative of this function, we get: 
d^2P/dt^2 = 0.44e^(1.6-0.2t) 
Since this is always positive, we know that the production function is concave up and there are no inflection points. Therefore, the point of diminishing returns occurs at the earliest possible time, which is when the worker has worked for 0 hours. So, the number of hours before the rate of production is maximized is 8-0 = 8. 
Thus, the answer is B) 11eb0d3b_9866_afbe_9790_abd54e853475_TB1243_11.

Suppose That the Total Number of Units Produced by a Worker