Question

If the garden is to be 1250 square feet, and the fence along the driveway costs $6 per foot while on the other three sides it costs only $2 per foot, find the dimensions that will minimize the cost.

Answer 1

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 $

Answer 2

So, the minimum cost is $400.

Area of the rectangle:

The area of a rectangle is the region occupied by a rectangle within its four sides or boundaries.

And the formula is,

$A=l\times b$

Given that,

Area of the garden=1250 square feet.

Let, the length be $x$ and the breadth be $y$ then,

$xy=1250...(1)$

The total cost of the fence is,

$C(x,y)=6x+2x+4y\\C(x,y)=8x+4y\\C(x)=8x+4(\frac{1250}{x} )$

Now, differentiating the obtained equation we get,

$C'(x)=8-\frac{4\times 1250}{x^2} =0\\x^2=625\\x=25\\y=50$

Therefore the length is 25 ft

And breadth is 50ft

Now, calculating the minimum cost,

$8(25)+4(50)=50\\=400$

Learn more about the area of the rectangle:

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