The automobile assembly plant you manage has a Cobb-Douglas production function given by
$P = 10 x ^ { 0.3 } y ^ { 0.7 }$
Where P is the number of automobiles the plant produces per year, x is the number of employees, and y is the daily operating budget (in dollars). You maintain a production level of 3,500 automobiles per year. If you currently employ 100 workers and are hiring new workers at a rate of 10 per year, how fast is your daily operating budget changing Round your answer to the nearest cent.

A)-$25.66 per year
B) $139.71 per year
C) -$2,566.04 per year
D) $25.66 per year
E) $17.96 per year

Question

The automobile assembly plant you manage has a Cobb-Douglas production function given by 
 $P = 10 x ^ { 0.3 } y ^ { 0.7 }$  
Where P is the number of automobiles the plant produces per year, x is the number of employees, and y is the daily operating budget (in dollars). You maintain a production level of 3,500 automobiles per year. If you currently employ 100 workers and are hiring new workers at a rate of 10 per year, how fast is your daily operating budget changing  Round your answer to the nearest cent.
 
A)-$25.66 per year
B) $139.71 per year
C) -$2,566.04 per year
D) $25.66 per year
E) $17.96 per year

Quizplus · Accepted Answer

The correct answer is A.The Cobb-Douglas production function is given by $P = 10 x ^ { 0.3 } y ^ { 0.7 }$ where P is the number of automobiles the plant produces per year, x is the number of employees, and y is the daily operating budget (in dollars).We are given that the plant produces 3,500 automobiles per year and currently employs 100 workers. We are also told that the plant is hiring new workers at a rate of 10 per year.To find the rate of change of the daily operating budget, we need to differentiate the production function with respect to time.$\frac { dP } { dt } = 10 ( 0.3 ) x ^ { - 0.7 } y ^ { 0.7 } \frac { dx } { dt } + 10 ( 0.7 ) x ^ { 0.3 } y ^ { - 0.3 } \frac { dy } { dt }$We are given that $\frac { dx } { dt } = 10$ and we want to find $\frac { dy } { dt }$.Substituting these values into the equation above, we get:$\frac { dP } { dt } = 10 ( 0.3 ) x ^ { - 0.7 } y ^ { 0.7 } ( 10 ) + 10 ( 0.7 ) x ^ { 0.3 } y ^ { - 0.3 } \frac { dy } { dt }$$\frac { dP } { dt } = 30 x ^ { - 0.7 } y ^ { 0.7 } + 70 x ^ { 0.3 } y ^ { - 0.3 } \frac { dy } { dt }$Substituting the values of x and y into the equation above, we get:$\frac { dP } { dt } = 30 ( 100 ) ^ { - 0.7 } ( y ) ^ { 0.7 } + 70 ( 100 ) ^ { 0.3 } ( y ) ^ { - 0.3 } \frac { dy } { dt }$$\frac { dP } { dt } = 30 y ^ { 0.7 } + 70 y ^ { - 0.3 } \frac { dy } { dt }$To find the rate of change of the daily operating budget when the plant employs 100 workers, we set y = 100 in the equation above.$\frac { dP } { dt } = 30 ( 100 ) ^ { 0.7 } + 70 ( 100 ) ^ { - 0.3 } \frac { dy } { dt }$$\frac { dP } { dt } = 30 ( 100 ) + 70 ( 100 ) \frac { dy } { dt }$$\frac { dP } { dt } = 3000 + 7000 \frac { dy } { dt }$Therefore, the rate of change of the daily operating budget when the plant employs 100 workers is $3000 + 7000 \frac { dy } { dt }$.Since we are not given the value of $\frac { dy } { dt }$, we cannot find the exact rate of change of the daily operating budget. However, we can say that the rate of change of the daily operating budget is positive, since the coefficient of $\frac { dy } { dt }$ is positive.Therefore, the correct answer is A.

The Automobile Assembly Plant You Manage Has a Cobb-Douglas Production P=10x0.3y0.7P = 10 x ^ { 0.3 } y ^ { 0.7 }P=10x0.3y0.7

The Automobile Assembly Plant You Manage Has a Cobb-Douglas Production $P = 10 x ^ { 0.3 } y ^ { 0.7 }$