Problem 48 - Intermediate Algebra

Question

A rectangular holding pen for cattle is to be designed so that its perimeter is 92 feet and its area is 525 feet. Find the dimensions of the holding pen.

Step-by-Step Solution

Answer

The dimensions are 25 feet by 21 feet.

Step 1: Identify the Variables

Let the length of the pen be \( l \) and the width be \( w \). These are the variables we need to find.

Step 2: Formulate the Equations

Using the perimeter and area, we have: \( 2l + 2w = 92 \) for the perimeter and \( lw = 525 \) for the area. We have two equations: 1. \( 2l + 2w = 92 \) 2. \( lw = 525 \).

Step 3: Simplify the Perimeter Equation

Divide the perimeter equation by 2 to make it simpler: \( l + w = 46 \).

Step 4: Express One Variable in Terms of the Other

From \( l + w = 46 \), express \( w \) in terms of \( l \): \( w = 46 - l \).

Step 5: Substitute into the Area Equation

Substitute \( w = 46 - l \) into the area equation \( lw = 525 \):\( l(46 - l) = 525 \). This simplifies to \( 46l - l^2 = 525 \).

Step 6: Rearrange the Equation

Re-arrange the equation to form a quadratic equation: \( l^2 - 46l + 525 = 0 \).

Step 7: Solve the Quadratic Equation

Use the quadratic formula \( l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -46, c = 525 \):\[ l = \frac{-(-46) \pm \sqrt{(-46)^2 - 4 \times 1 \times 525}}{2 \times 1} \]\[ l = \frac{46 \pm \sqrt{2116 - 2100}}{2} \]\[ l = \frac{46 \pm \sqrt{16}}{2} \]\[ l = \frac{46 \pm 4}{2} \].

Step 8: Calculate the Possible Lengths

The possible solutions for \( l \) are:\( l = \frac{46 + 4}{2} = 25 \) and \( l = \frac{46 - 4}{2} = 21 \).

Step 9: Find Corresponding Widths

For \( l = 25 \), \( w = 46 - 25 = 21 \).For \( l = 21 \), \( w = 46 - 21 = 25 \).So, the dimensions are either \( 25 \times 21 \) or \( 21 \times 25 \).

Key Concepts

Understanding Area and Perimeter

When working with geometric figures like rectangles, two important measurements come into play: area and perimeter.
The perimeter of a rectangle is the total distance around the outer edge. It is calculated by the formula

• Perimeter = 2(length + width).

For our cattle pen example, the perimeter is set at 92 feet. This gives us one equation to work with.

The area of a rectangle, on the other hand, is the measure of how much surface the shape covers. The area is given by the formula:

• Area = length × width.

In this case, the area of the cattle pen is 525 square feet.

Understanding these formulas helps us establish equations that represent the physical constraints of geometric shapes. In this exercise, these constraints help us solve for the rectangle's dimensions, ensuring it matches the required specifications.

Applying the Quadratic Formula

The quadratic formula provides a way to solve equations of the form

• \[ ax^2 + bx + c = 0 \]

where \( a \), \( b \), and \( c \) are constants.

In our problem, once we express the length and width of the rectangle in terms of a single variable, we end up with a quadratic equation:

• \[ l^2 - 46l + 525 = 0 \]

To solve this, we use the quadratic formula:

• \[ l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 1 \), \( b = -46 \), and \( c = 525 \). By substituting these values into the formula, we find potential values for \( l \), either 25 or 21.

This powerful formula is especially useful in geometry problems, allowing you to find unknown values even when they initially seem difficult to determine.

Solving Geometry Problems

Geometry problems often require a mix of algebraic and geometric reasoning. In this example, we needed to find the dimensions of a holding pen by setting up equations based on area and perimeter.

To effectively tackle these problems:

• Identify what you need to find and assign variables.
• Formulate equations based on geometric principles like area and perimeter.
• Simplify and manipulate the equations to isolate unknowns.
• When needed, apply algebraic methods such as the quadratic formula to solve for variables.

By following these steps, you can solve various kinds of geometry problems.

These problems not only test your mathematical skills but also improve your logical reasoning and problem-solving abilities. They show the importance of integrating different areas of math to find solutions to real-world problems.