Questions: A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a straight river. He does not need a fence along the river. What are (a) Experiment with the problem by drawing several diagrams illustrating the situation. Calculate the area of each configuration, and use your results to estimate the comma-separated list.) (b) Find a function that models the area of the field in terms of one of its sides. A(x)=

A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a straight river. He does not need a fence along the river. What are (a) Experiment with the problem by drawing several diagrams illustrating the situation. Calculate the area of each configuration, and use your results to estimate the comma-separated list.) (b) Find a function that models the area of the field in terms of one of its sides. A(x)=

Transcript text: A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a straight river. He does not need a fence along the river. What are (a) Experiment with the problem by drawing several diagrams illustrating the situation. Calculate the area of each configuration, and use your results to estimate the comma-separated list.) (b) Find a function that models the area of the field in terms of one of its sides. \[ A(x)=\square \]

Solution

Solution Steps

Step 1: Understanding the Problem

The farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a straight river. The river side does not need fencing. We need to find the dimensions that maximize the area of the field.

Step 2: Define Variables

Let:

\( x \) be the length of the field parallel to the river.
\( y \) be the width of the field perpendicular to the river.

Step 3: Express the Perimeter Constraint

The total length of the fencing used is given by: \[ 2y + x = 2400 \]

Step 4: Express the Area Function

The area \( A \) of the rectangular field is: \[ A = x \cdot y \]

Step 5: Solve for \( y \) in Terms of \( x \)

From the perimeter constraint: \[ y = \frac{2400 - x}{2} \]

Step 6: Substitute \( y \) into the Area Function

Substitute \( y \) into the area function: \[ A(x) = x \cdot \frac{2400 - x}{2} \] \[ A(x) = \frac{2400x - x^2}{2} \] \[ A(x) = 1200x - \frac{x^2}{2} \]