Questions: A fence is to be built to enclose a rectangular area of 1800 square feet. The fence along three sides is to be made of material that costs 3 per foot. The material for the fourth side costs 9 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.

A fence is to be built to enclose a rectangular area of 1800 square feet. The fence along three sides is to be made of material that costs 3 per foot. The material for the fourth side costs 9 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.

Transcript text: A fence is to be built to enclose a rectangular area of 1800 square feet. The fence along three sides is to be made of material that costs $3 per foot. The material for the fourth side costs $9 per foot. Find the dimensions of the rectangle that will allow for the most economical fence to be built. The short side is $\square$ ft and the long side is $\square$ ft.

Solution

Solution Steps

Step 1: Express the cost of the fence in terms of the dimensions of the rectangle

The total cost $C$ is given by $C = 2C_1w + C_1l + C_2l$, where $l$ is the length and $w$ is the width of the rectangle.

Step 2: Use the constraint that the area $A = l \times w$ to eliminate one variable

Express $l$ in terms of $w$ and $A$: $l = \frac{A}{w}$.

Step 3: Substitute $l$ from step 2 into the cost equation from step 1 to get the cost in terms of a single variable, $w$

After substitution, the cost function in terms of $w$ becomes $C = 2C_1w + (C_1 + C_2)\frac1800w$.

Step 4: Find the derivative of the cost function with respect to $w$ and set it to zero to find the critical points

The derivative of the cost function with respect to $w$ is $6 - \frac{21600}{w^{2}}$. Setting this to zero gives the critical points.

Step 5: Determine the dimensions $l$ and $w$ that minimize the cost

The critical point that minimizes the cost is $w = 60$ feet (rounded to 2 decimal places). Substituting this value into $l = \frac1800w$ gives $l = 30$ feet (rounded to 2 decimal places).