Problem 27 - Algebra and Trigonometry

Question

Length of a Garden \(\quad\) A rectangular garden is 25 \(\mathrm{ft}\) wide. If its area is \(1125 \mathrm{ft}^{2},\) what is the length of the garden?

Step-by-Step Solution

Answer

The length of the garden is 45 ft.

Step 1: Understand the Problem

You are given the width of the rectangular garden as 25 ft and the area of the garden as 1125 sq ft. You need to find the length of this garden.

Step 2: Recall the Formula for Area of a Rectangle

The area of a rectangle can be calculated using the formula: \( \text{Area} = \text{width} \times \text{length} \).

Step 3: Set Up the Equation

Using the given data, substitute the known values into the area formula: \( 1125 = 25 \times \text{length} \).

Step 4: Solve for Length

To find the length, divide both sides of the equation by 25: \( \text{length} = \frac{1125}{25} \).

Step 5: Calculate the Length

Perform the division: \( \text{length} = 45 \). So, the length of the garden is 45 ft.

Key Concepts

Area of a Rectangle

The concept of the area of a rectangle is fundamental in geometry and important in solving various practical problems. The area represents the amount of space a rectangle occupies on a plane.
The formula to calculate this is straightforward:

• Area = width x length

For our rectangular garden problem, the width is given as 25 ft, and the area is provided as 1125 square feet.
Our task is to compute one of the missing dimensions using what we know. When dealing with rectangles, always remember that the area is a product of its width and length.
This simple concept helps in various fields including architecture, gardening, and interior design, where knowing how much space something takes up is essential.

Solving Equations

Solving equations is a fundamental algebraic skill that allows us to find unknown values. In the context of the rectangular garden problem, our unknown is the length of the garden.
The equation we use derives from the area formula:

• Given: Area = width x length, substitute: 1125 = 25 x length

To find the length, we need to isolate it on one side of the equation.Perform this by dividing both sides by the known width (25):

• length = \( \frac{1125}{25} \)

This approach—solving for an unknown by simplifying the equation—demonstrates how equations are powerful tools for finding missing information.
Understanding how to manipulate and solve equations enables you to tackle a variety of mathematical challenges in everyday life.

Geometry Problem Solving

Geometry problem solving involves using known formulas and logical reasoning to find missing information. In this problem, we use simple geometry concepts and algebra.
The given area and width allow us to determine the unknown length of the garden. Here’s a structured approach to tackle such geometry problems:

• Understand what is known and what is unknown.
• Recall relevant formulas, such as those for perimeter, area, or volume, depending on the problem.
• Formulate an equation using the known values and solve it to find the unknown.

Breaking down problems into these steps not only helps in solving them but also in verifying the answer for correctness.
This methodical approach ensures that every piece of information is used effectively, which is crucial for tackling more complex geometry problems in the future.