A landscape architect wishes to enclose a rectangular garden on one side by a stone wall that costs 50 dollars per foot and on the other three sides by a wood fence that costs 15 dollars per foot.

Question

A landscape architect wishes to enclose a rectangular garden on one side by a stone wall that costs 50 dollars per foot and on the other three sides by a wood fence that costs 15 dollars per foot. The side with the stone wall must be between 10 feet and 50 feet. If the area of the garden is 675 square feet, find the dimensions of the garden that minimize the cost.

Use the following variables: A=area of the garden; C=total cost of the wall and fence; x=length of the stone wall (and opposite side); y=length of the other two sides.

(1) Find a formula for the area A in terms of x and y.

A =  

(2) Find a formula for the cost C in terms of x and y.

C =  

(3) Find a formula for the cost C as a function of x alone.

C(x)=  

(4) Find the derivative C′(x).

C′(x)=  

(5) Solve the equation C′(x)=0 for x.

[Give only the answer that is in the domain: 10<x<50]

x =  

(6) Find the minimum cost.

Minimum cost =  

Answer 1

A=675=xy

y=675/x

C=50x+15x+15(2y)

C=65x +30(675/x)

C= 65x +20250/x

C'=65-20250/x^2=0

65x^2=20250

x^2=20250/65

x=sqr(20250/65)= about 17.65 feet

minimum cost =65x+20250/x=65sqr(20250/65)+ 20250/sqr(20250/65)

=about $2,294.56

rounded up to nearest cent

Answer 2

(1) For area of a rectangle,

A = x * y = 675

(2) C = 50*x + 15*x + 2 * 15*y = 65*x + 30*y

(3) Rearrange (1) and substitute for y into (2),

y = 675/x

C = 50*x + 15*x + 2*15* 675/x = 65*x + 20250/x

(4) C'(x) = 65 + 20250* (-1/x²) = 65 - 20250/x²

(5) For "minimum" cost, C' = 0, therefore

0 = 65 - 20250/x²

20250/x² = 65

x² = 20250/65 = 311.538

x = sqrt(311.538) = 17.650 ft

y = 675/x = 675/17.650 = 38.243 ft

(6) Using (2)

C_min = 65*17.650 + 30*38.243

C_min = 1147.25 + 1147.25 = 2294.5

C_min = $2294.50

Answer 3

xy=675 => x=675/y

C=(50*675/y) + 15y

dC/dy=(-50*675/y^2)+15=0

Solve for y then plug in for x...and in fact the limitation of x will be met.