Can i please get help with business calculus math homework?

Question

Application of the Derivative: 2.6 #14

 

A rectangular corral of 54 square meters is to be fenced off te divided by a fence into two sections, if the cost of the fencing for the boundary is $5 per meter and the dividing fence costs $2 per meter, find the dimensions of the corral tht minimize the cost of the fencing

Answer 1

Let x = length of fence that divides the two corrals

     y = length of a side of the fence perpendicular to the dividing fence

 

Then, xy = 54

 

C = cost of fence = 5(2x+2y) + 2x

 

So, C = 12x + 10y

 

Since xy = 54, we have y = 54/x

 

So, C = 12x + 540/x

 

C' = 12 - 540/x²

 

    = (12x² - 540) / x²

 

C' = 0 if 12x² = 540

 

                x² = 45

 

                x = √45 = 3√5 m 

 

When 0 < x < 3√5, C' < 0  (so C is decreasing)

When x > 3√5, C' > 0 (so C is increasing)

 

Therefore, C is minimized when x = 3√5 m and y = 54/(3√5) = 18√5/5 m.