Question

Question 2 options: A fence must be built to enclose a rectangular area of 45,000 ft2. Fencing material costs $4 per foot for the two sides facing north and south and $8 per foot for the other two sides. Find the length and width that will produce the least expensive fence.

Answer 1

The length that would produce the least expenses is
`A= 300 ft`

The width that would produce the least expenses is
`B= 150 ft`

Explanation:

From the question we are told that the

Required area to enclose is
`A = 45,000 ft^2`

Fencing material cost is
`C =`$4 per foot for north and south

Fencing material cost for east and west
`C_(E/W) =` $8

The diagram for this question is shown on the first uploaded image

From the diagram is mathematically evaluated as

AB = 45000

=>
`B = (45000)/(A)`

The overall cost of building this fence is

`T = 2 A (4) + 2(B)(8)`

Substituting for B in the equation above

`T = 8A +16 ((45000)/(A) )`

differentiating both sides with respect to x

`T' (A) = 8 - (16 *45000)/(A^2)`

At minimum possible cost
`T'(A) = 0`

=>
`8 - (16 *45000)/(A^2) = 0`

`8A^2 = 16*45000`

`A^2 = (16*45000)/(8)`

`A = \sqrt{( 16*45000)/(8) }`

`= 300 ft`

Then B is mathematically evaluate as

`B = (45000)/(300)`

`= 150 ft`

Then the maximum is mathematically evaluated as

`T = 8 (300) + 16(150)`

=$4800