Question

A rectangualr garden is to be enclosed usinf two different types of fencinf. one side of the garden boarders a road and the fencing for this side costs $25 per foot. The fencinf for the other sides costs $10 per foot.

Answer 1

x = 18.9737 ft

y = 15.8114 ft

Explanation:

Given:

- Area of rectangle garden A = 300 ft^2

- Cost of one side fencing = $25 /ft

- Cost of second side fencing = $10 /ft

Find:

- Find the dimensions to minimize the cost.

Solution:

- Develop a total cost function for the rectangular garden as follows:

Total Cost = x*25 + 3*y*10

T.C = 25*x + 30*y

- Use the area expression to find the relationship between x and y dimensions:

A = x*y

300 = x*y

y = 300 / x

- Input the above relationship into the total cost function:

T.C = 25*x + 30*300/x

T.C = 25*x + 9000/x

- To minimize the function, take its derivative with respect to dimension x:

dC/dx = 25 - 9000/x^2

- Set derivative to 0:

0 = 25 - 9000/x^2

Solve for x:

0 = 25x^2 - 9000

x = sqrt(9000/25)

x = 18.9737 ft

- The corresponding dimension y is:

y = 300/18.9737 = 15.8114 ft