Question

Celeste wants to build a fence to enclose a rectangular area of 600 square feet near a stream. The side of the enclosure next to the stream will be made of a waterproof fencing. The other three sides will be made of wood fencing. The waterproof fencing costs $20 per foot, and the wood fencing costs $10 per foot. What is the minimum total cost for the fencing?

Answer 1

Answer:

Minimum cost of fencing is $1200.

Step-by-step explanation:

Let the length of the sides parallel to the stream = l feet

And width of the rectangular area = w

Area of the rectangle = lw = 600 square feet

l = $\frac{600}{w}$------(1)

Since cost of the fencing on the stream side = $20 per foot

So to cover the side with waterproof fencing = 20l

Similarly to cover all three sides of the rectangle with wooden fence

= 10(l + 2w)

Total cost C for fencing = 10(l + 2w) + 20l

C = 10l + 20w + 20l

C = 30l + 20w

Now replace the value of l from equation (1)

$C=30(\frac{600}{w})+20w$

$C=(\frac{18000}{w})+20w$

To minimize the cost we will find the derivative of C with respect to w and then equate it to zero.

$\frac{dC}{dw}=\frac{d}{dw}[(\frac{18000}{w})+20w]$

$C'=-\frac{18000}{w^{2}}+20$

C' = 0

$0=-\frac{18000}{w^{2}}+20$

$\frac{18000}{w^{2}}=20$

w² = $\frac{18000}{20}$

w = $\sqrt{900}$

w = 30 feet

Now replace the value of w in the expression of total cost 'C'

$C=(\frac{18000}{30})+20(30)$

C = 600+ 600

C = $1200  

Therefore, the minimum cost of the fencing is $1200.