Question

Bob wants to create two pens, as shown in the figure. One pen if for a garden and it needs a heavy duty fence to keep out the critters. This heavy duty fence costs $15 per foot. The dog pen shares a side with the garden and has a lighter weight fence on the other three sides that costs $5 per foot. If each pen is to have an area of 504, find the values of x and y that would minimize the total cost of the fencing.

Answer 1

you got a picture?

Answer 2

looks like we got \(x^2=504\) and \(xy=504\) cost as i read it is
\[4x\times 15+5x+2yx\]

Answer 3

ooop lets try that again
cost is
\[4x\times 15+5x+2y\times 5=60x+5x+10y=65x+10y\]

Answer 4

to make this a function only of \(x\) note that
\[xy=504\] so
\[y=\frac{504}{x}\] now your cost is
\[C(x)=65x+\frac{5040}{x}\] and you can minimize that one

Answer 5

so i find the derivative of the cost function, find x and then plug that back into xy=504 to find y?

Answer 6

yes
assuming you mean
find the derivative, set it equal to zero and solve, then plug that back in to find y

Answer 7

c'(x)=65-5040x^-2
if you set that equal to zero and solve x=\[12\sqrt{7/13}\]
so i would take 504 and divide by x?

Answer 8

actually it would be
30x+10x+5y+30y = 40x +35y
which means your cost function is
C(x) = 40x+17640/x
then taking the derivative
C'(x) = 40-17640x^-2
set that equal to zero and solve
x=21
so 504 = 21y
y =24
those are the correct answers for x and y