Question

a fence is to be built to enclose a rectangular area of 800 square feet. the fence along three sides is to be made of material that costs $5 per foot. the material for the fourth side costs $15 per foot. find the dimensions of the rectangle that will allow for the most economical fence to be built. the short side is □ ft and the long side is □ ft.

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 \( 10-\frac{16000}{x^{2}}=0\).

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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.