Answer:
Dimensions of rectangular garden:
x = 25 feet ( sides along the driveway)
y = 50 feet
Step-by-step explanation:
Rectangular area is:
A(r) = x*y (1)
if we call x one the driveway side the cost of that side will be
6*x
The cost of the other side parallel to driveway side is 2*x and cost of the others two sides are 4*y
Total costs: C = 6*x + 2*x * 4*y (2)
From equation (1)
A(r) = 1250 = x*y ⇒⇒ y = 1250/ x
Plugging that value in equation (2) we get costs as a function of x
that is:
C(x) = 6*x + 2*x + 4* 1250/x
Taking derivatives on both sides of the equation
C´(x) = 6 + 2 - 5000/x²
C´(x) = 8 - 5000 /x²
C´(x) = 0 ⇒ 8 - 5000 /x² = 0
8*x² -5000 = 0
x² = 5000/8
x² = 625
x = 25 feet
and y = 1250/ 25
y = 50 ft
C(min) = 50*2*2 + 6*25 + 2*25
C(min) = 200 + 200
C(min) = 400 $
