a fence is to be built to enclose a rectangular area of 800 square feet. the fence along three sides is to…

Explanation:

Step1: Let the short side be ( x ) feet

Let the short side be ( x ) feet. Since the area ( A = 800) square feet and ( A=\text{length}\times\text{width}), the long side ( y=\frac{800}{x}) feet.

Step2: Write the cost function

The cost function ( C): The three - side fence (two short sides and one long side) cost ( 5) per foot and the fourth side (the other long side) cost ( 15) per foot. So ( C = 5(2x)+5y + 15y=10x + 20y). Substitute ( y=\frac{800}{x}) into the cost function: ( C(x)=10x+20\times\frac{800}{x}=10x+\frac{16000}{x}), ( x>0).

Step3: Find the derivative of the cost function

Differentiate ( C(x)) with respect to ( x). Using the power rule ((x^n)^\prime=nx^{n - 1}), ( C^\prime(x)=10-\frac{16000}{x^{2}}).

Step4: Set the derivative equal to zero

Set ( C^\prime(x) = 0), then ( 10-\frac{16000}{x^{2}}=0). [ \begin{align*} 10&=\frac{16000}{x^{2}}\ x^{2}& = 1600\ x&=40 \end{align*} ]

Step5: Check the second - derivative

Differentiate ( C^\prime(x)) to get ( C^{\prime\prime}(x)=\frac{32000}{x^{3}}). When ( x = 40), ( C^{\prime\prime}(40)=\frac{32000}{40^{3}}=\frac{32000}{64000}=0.5>0). So the cost function is minimized at ( x = 40).

Step6: Find the long side

When ( x = 40), ( y=\frac{800}{40}=20).

Answer:

The short side is ( 20) ft and the long side is ( 40) ft.