Question

Arectangularplotoflandistobefencedinusingtwokinds of fencing. Two opposite sides will use heavy-duty fencing selling for $3 a foot, while the remaining two sides will use standard fencing selling for $2 a foot. What are the di- mensions of the rectangular plot of greatest area that can be fenced in at a cost of $6000?

Answer 1

Dimensions of the rectangular plot will be 500 ft by 750 ft.

Explanation:

Let the length of the rectangular plot = x ft.

and the width of the plot = y ft.

Cost to fence the length at the cost $3.00 per feet = 3x

Cost to fence the width of the cost $2.00 per feet = 2y

Total cost to fence all sides of rectangular plot = 2(3x + 2y)

2(3x + 2y) = 6,000

3x + 2y = 3,000 ----------(1)

3x + 2y = 3000

2y = 3000 - 3x

y =
`(1)/(2)[3000-3x]`

y = 1500 -
`(3x)/(2)`

Now area of the rectangle A = xy square feet

A = x[
`1500-(3x)/(2)`]

For maximum area
`(dA)/(dx)=0`

A' =
`(d)/(dx)(1500x-(3)/(2)x^(2))` = 0

1500 - 3x = 0

3x = 1500

x = 500 ft

From equation (1),

y = 1500 -
`(3)/(2)* 500`

y = 1500 - 750

y = 750 ft

Therefore, for the maximum area of the rectangular plot will be 500 ft × 750 ft.

two fencing 3(500+500) = $3000

other two fencing 2(750+750) = $3000