Calculus

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A rectangular garden of area 75 sq ft is to be surrounded on 3 sides by a brick wall costing \$12 per foot and on one side by a fence costing \$7 per foot. Find the dimensions of the garden that the cost of the materials is minimized Read more: A rectangular garden of area 75 sq ft is to be surrounded on 3 sides by a brick wall costing \$10 per foot and on one side by a fence costing \$5 per foot. Find the dimensions of the garden that the cost of the materials is minimized

• Calculus -

Let the dimensions of the garden be
x (fence side) and 75/x (since the area is 75).
So the total cost, C(x)
= 7x+2*12(75/x)+12(7)
= 7x+1800/x+84
To find the minimum cost (at the expense of the shape of the garden)
we calculate
C'(x)=0
7-1800/x^2=0
x=sqrt(1800/7)=16.04'
75/x=4.68'

• Calculus -

• Calculus -

let the side with the fence be x ft long, then the perpendicular side is 75/x

cost = 7x + 12(75/x)
d(cost)/dx = 7 - 900/x^2 = 0 for a min of cost
7 = 900/x^2
x^2 = 900/7
x = 30/√7 = appr 11.339

so the garden has to be 11.339 by 75/11.339
or 11.34 by 6.61

check: 11.34 x 6.61 = 74.96 , not bad
cost = 7(11.339) + 900/11.339 = 158.75 --- minimum cost

let x = 11 , cost = 7(11) + 900/11 = 158.82 , a higher cost
let x = 11.6, cost = 7(11.5) + 900/11.5 = 158.75 -- just a bit higher

• Calculus -

You're right, I mis-calculated the cost of the fourth side (12(7) instead of 12x).

Let the dimensions of the garden be
x (fence side) and 75/x (since the area is 75).

Since there are four sides to the garden:

So the total cost, C(x)
= 7x+2*12(75/x)+12(x)
= 19x+1800/x
To find the minimum cost (at the expense of the shape of the garden)
we calculate
C'(x)=0
19-1800/x^2=0
x=sqrt(1800/19)=9.733' (fence side)
75/x=7.706' (wall side)
C(sqrt(1800/19))=\$369.86