Question

The automobile assembly plant you manage has a Cobb-Douglas production function given by
P = (10x^0.3)(y^0.7)
where P is the number of automobiles it produces per year, x is the number of employees, and y is the daily operating budget (in dollars). You maintain a production level of 600 automobiles per year. If you currently employ 150 workers and are hiring new workers at a rate of 5 per year, how fast is your daily operating budget changing?

Answer 1

Is this calculus?

Answer 2

yes

Answer 3

Differentiate both sides with respect to \(t\) using the chain rule.
Note that \(P' = 0\) since production remains constant. \(x' = 5\).

Answer 4

You're going to have to use the equation pre-differentiation to figure out the value of \(y\).
Your goal is to find \(y'\)

Answer 5

how do I do that?

Answer 6

@wio

Answer 7

We need to use this equation to find \(y\) first of all.\[
P = (10x^{0.3})(y^{0.7})
\]

Answer 8

"You maintain a production level of 600 automobiles per year"\[
P = 6000
\]"you currently employ 150 workers"\[
x=150
\]

Answer 9

So this is enough info to solve for \(y\)

Answer 10

@krizzym You don't know how to plug and chug this thing?

Answer 11

@wio I'm trying to learn how to do it and i wanted to see the steps to getting the answer

Answer 12

@wio what do I do with the 0.7 so i can get y by itself?

Answer 13

\(0.7 = 7/10\) So the reciprocal is \(10/7\). Raise both sides to that power.

Answer 14

@wio Im still not getting it

Answer 15

@wio can you show me a step by step review?

Answer 16

http://www.wolframalpha.com/input/?i=6000++%3D+%2810%28150%29%5E0.3%29%28y%5E0.7%29

Answer 17

\[\begin{array}{rcl}
6000 &=& (10(150)^{0.3})(y^{0.7}) \\
\frac{600}{150^{0.3}} &=& y^{0.7} \\
\left(\frac{600}{150^{0.3}}\right)^{1/0.7} &=& y \\

\end{array}
\]

Answer 18

Can you follow this?

Answer 19

@yes

Answer 20

I hope there is no user named yes.

Answer 21

sorry i meant to @ your username @wio lol thanks

Answer 22

\[
P = (10x^{0.3})(y^{0.7})
\]We differentiate both sides with respect to \(t\). We have to use chain rule: \[
P' = (10x^{-0.7}x')(y^{-0.3}y')
\]We know \(P'=0,x=150,x'=5\)

Answer 23

"You maintain a production level of 600 automobiles per year."
\[
P'=0
\]
" hiring new workers at a rate of 5 per year" \[
x'=5
\]

Answer 24

I gotta go, good luck.

Answer 25

@wio ok thanks