Question

A rectangular plot of land is to be fenced in using two types of fencing. Two opposite sides will use heavy-duty fencing selling for $4.50 a foot. The two remaining sides will use standard fencing selling for $3 a foot. How much of the heavy-duty and standard fencing should be used so that the greatest area can be fenced in at a cost of $18,000?

Answer 1

The amount of heavy-duty fencing to be used is 2000 ft for $9000 and the amount of standard fencing to be used is 3000 ft for $9000 in order to make a total cost of $18000.

How do we determine the area of a rectangular plot?

The area of the rectangular plot is denoted by the length multiplied by the width.

Let us assume that the length = x and the width = y.

• Area = xy

Then, for two opposite with heavy fencing for $4.50 a foot and the standard fencing for $3 a foot, we have:

2x (4.50) + 2y(3) = 18000  ---- (1)

⇒ 9x + 6y = 18000

Make (y) the subject, we have:

$\mathbf{y = \dfrac{18000-9x}{6}}$

Replace the value of y with the area of the rectangle, and we get:

$\mathbf{Area =x(\dfrac{18000 - 9x}{6})}$

Area = x(3000 - 1.5x)

Area = 3000x - 1.5x²

For the area to be maximum, we take the differentiation of the Area:

dA/dx = 0

dA/dx = 3000 - 3x

x = 3000/3

x = 1000 ft

From equation (1)

2x (4.50) + 2y(3) = 18000  

x (4.50) + y(3) = 9000  

1000(4.50) + 3y = 9000

3y = 9000 -4500

y = 4500/3

y = 1500 ft

So, there are (1000× 2)ft = 2000ft heavy duty fencing for $9000, and (1500 ×2)ft = 3000ft standard fencing for $9000 to make a cost of $18000.

Learn more about calculating the area of a rectangular plot here:

brainly.com/question/21297672

#SPJ1