Question

solve the following problems. draw a diagram for each problem (2pts.). label your primary equation and if needed secondary equation (2pts.). show your work(4 pts.). make sure you put the units of measuremtn in your answer (1pt).

1. a rancher wants to construct two identical rectangular corrals using 500 ft. of fencing. the rancher decides to build them adjacent to each other, so they share fencing on one side. what dimensions should the rancher use to construct each corral so that together, they will enclose the largest possible area?

Explanation:

Step1: Define variables and primary equation

Let the length of each corral parallel to the shared side be \(x\) (in feet) and the width perpendicular to the shared side be \(y\) (in feet).
The total fencing used is \(3x + 4y=500\) (primary equation). We can express \(y=\frac{500 - 3x}{4}\).
The area \(A\) of the two - corral combination is \(A = 2xy\) (secondary equation). Substitute \(y\) into the area formula: \(A(x)=2x\times\frac{500 - 3x}{4}=\frac{1}{2}(500x-3x^{2})\).

Step2: Find the derivative of the area function

Differentiate \(A(x)\) with respect to \(x\). Using the power rule \((x^n)^\prime=nx^{n - 1}\), \(A^\prime(x)=\frac{1}{2}(500 - 6x)\).

Step3: Set the derivative equal to zero and solve for \(x\)

Set \(A^\prime(x)=0\), so \(\frac{1}{2}(500 - 6x)=0\).
Multiply both sides by \(2\): \(500-6x = 0\).
Then \(6x=500\), and \(x=\frac{500}{6}=\frac{250}{3}\approx83.33\) ft.

Step4: Find \(y\)

Substitute \(x = \frac{250}{3}\) into \(y=\frac{500 - 3x}{4}\).
\(y=\frac{500-3\times\frac{250}{3}}{4}=\frac{500 - 250}{4}=\frac{250}{4}=62.5\) ft.

Answer:

Each corral should have dimensions \(x=\frac{250}{3}\text{ ft}\) (length parallel to the shared side) and \(y = 62.5\text{ ft}\) (width perpendicular to the shared side).