Applied Min/Max

Question

You have T dollars to buy fence to enclose a rectangular plot of land. The fence for the top and bottom costs $5 per foot and for the sides it costs $3 per foot. Find the dimensions of the plot with the largest area. For this largest plot, how much money was used for the top and bottom, x, and for the sides, y?

Answer 1

I'm assuming you mean that x is the length of each of the top and bottom, and y is the length of each side.

 

The perimeter, which is the length of the fence, is 2x + 2y.  The total cost of the fence, then, is 2x * 5 + 2y * 3 = 10x + 6y.

 

You know that this is equal to T, so we can find an expression for y in terms of x:

 

10x + 6y = T

6y = T - 10x

y = (T - 10x) / 6

 

So the area of the plot is A = xy = x (T - 10x) / 6 = (Tx - 10x²) / 6

 

So let's maximize this:

 

dA/dx = D_x (Tx - 10x²) / 6 = T/6 - 20x/6

 

Setting this equal to 0 gives:

 

T/6 - 20x/6 = 0

T - 20x = 0

T = 20x

T/20 = x

 

Now figure out y:

y = (T - 10x) / 6 =

(T - 10 * T/20) / 6 =

(T - T/2) / 6 =

T/2 / 6 =

T/12

 

So as a function of T, the amount of money you have to start with, the dimension of the plot will be T/20 (top and bottom) and T/12 (sides).

 

The top and bottom will cost T/20 * 5 = T/4 dollars each.  And the sides will cost T/12 * 3 = T/4 dollars each.

 

So each side of the rectangle will cost T/4 dollars, adding up to a total of T, as expected.