Answer:
x = 18.9737 ft
y = 15.8114 ft
Step-by-step 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
