Dimensions of the garden that minimize the cost

Question

A landscape architect wished to enclose a rectangular garden on one side by a brick wall costing $60/ft and on the other three sides by a metal fence costing $30/ft. If the area of the garden is 122 square feet, find the dimensions of the garden that minimize the cost.

 

length of side with bricks x 

length of adjacent side y

Answer 1

the cost of the fence is 60x + 30x + 2*30y = 90x + 60y

Also, we know that xy = 122

Substituting:  cost = 90x + 60 (122/x) = 90x + 7320/x

 

The minimum and maximum of a function occur where its first derivative equals zero, infinity, or is undefined.

The first derivative of this function is 90 - 7320/x^2.

 

Set this equal to zero:  90 = 7320/x^2, or x^2 = 7320/90 = 81.33333

 

Then x is the square root of 81-1/3 and y is 122/x