Questions: Complete Section 3.5 Activity A lifeguard needs to rope off a rectangular swimming area in front of Long Lake Beach, using 500 yd of rope. What dimensions of the rectangle will maximize the area? (Note that one side of the rectangle is the shoreline.) Let x be the length of a side of the rectangle perpendicular to the shoreline. Write the objective function for the area in terms of x. A(x) = 500x - 2x^2 (Type an expression using x as the variable.) The length of the shorter side of the rectangular region is [ ] y The length of the longer side of the rectangular region is [ ] y

Complete Section 3.5 Activity

A lifeguard needs to rope off a rectangular swimming area in front of Long Lake Beach, using 500 yd of rope. What dimensions of the rectangle will maximize the area? (Note that one side of the rectangle is the shoreline.)

Let x be the length of a side of the rectangle perpendicular to the shoreline. Write the objective function for the area in terms of x.

A(x) = 500x - 2x^2
(Type an expression using x as the variable.)

The length of the shorter side of the rectangular region is [ ] y
The length of the longer side of the rectangular region is [ ] y

Transcript text: Complete Section 3.5 Activity A lifeguard needs to rope off a rectangular swimming area in front of Long Lake Beach, using 500 yd of rope. What dimensions of the rectangle will maximize the area? (Note that one side of the rectangle is the shoreline.) Let x be the length of a side of the rectangle perpendicular to the shoreline. Write the objective function for the area in terms of x. A(x) = 500x - 2x^2 (Type an expression using x as the variable.) The length of the shorter side of the rectangular region is [ ] y The length of the longer side of the rectangular region is [ ] y

Solution

Solution Steps

To maximize the area of a rectangular swimming area with one side along the shoreline, we need to express the area as a function of one variable and then find the maximum value of this function. Given that the total length of the rope is 500 yards, and one side is along the shoreline, the perimeter constraint can be expressed as \( x + 2y = 500 \), where \( x \) is the length perpendicular to the shoreline and \( y \) is the length parallel to the shoreline. The area \( A \) is given by \( A = x \times y \). We can express \( y \) in terms of \( x \) using the perimeter constraint and then substitute it into the area formula to get a function of \( x \) only. Finally, we find the value of \( x \) that maximizes this area function.

Step 1: Define the Variables

Let \( x \) be the length of the side of the rectangle perpendicular to the shoreline. The length of the side parallel to the shoreline, denoted as \( y \), can be expressed in terms of \( x \) using the perimeter constraint: \[ x + 2y = 500 \implies y = \frac{500 - x}{2} \]

Step 2: Express the Area Function

The area \( A \) of the rectangle can be expressed as: \[ A = x \cdot y = x \left(\frac{500 - x}{2}\right) = \frac{500x - x^2}{2} \]

Step 3: Find the Critical Points

To maximize the area, we take the derivative of \( A \) with respect to \( x \) and set it to zero: \[ \frac{dA}{dx} = \frac{500 - 2x}{2} = 0 \implies 500 - 2x = 0 \implies x = 250 \]

Step 4: Calculate the Dimensions

Substituting \( x = 250 \) back into the equation for \( y \): \[ y = \frac{500 - 250}{2} = \frac{250}{2} = 125 \]

Step 5: Verify the Maximum Area

The maximum area can be calculated as: \[ A = x \cdot y = 250 \cdot 125 = 31250 \]