Problem 35 - Algebra and Trigonometry

Question

Length of a Garden 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 feet.

Step 1: Understand the Problem

We are given a rectangular garden with a width of 25 feet and an area of 1125 square feet. We need to find the length of the garden.

Step 2: Write the Formula for Area of a Rectangle

The area of a rectangle is given by the formula: \[ \text{Area} = \text{Length} \times \text{Width} \] In this problem, the area is 1125 square feet, and the width is 25 feet.

Step 3: Rearrange the Formula to Find Length

Rearrange the formula to solve for the length: \[ \text{Length} = \frac{\text{Area}}{\text{Width}} \]

Step 4: Substitute the Known Values

Substitute the known values into the formula: \[ \text{Length} = \frac{1125}{25} \]

Step 5: Calculate the Length

Divide 1125 by 25 to find the length of the garden: \[ \text{Length} = 45 \]

Step 6: Solution Statement

The length of the garden is 45 feet.

Key Concepts

rectangular garden

A rectangular garden offers a simple and clear example of how geometric principles apply to real-world situations. The shape of the garden is a rectangle, which is a four-sided figure with opposite sides that are equal and parallel. When we describe this garden, the two pivotal dimensions are its width and length.
In practical terms, knowing the dimensions of your garden helps you plan how to lay out plants and pathways, ensuring every inch is used effectively.
Understanding the size of each plot is vital for gardening and landscape planning. Utilizing rectangles in garden planning makes it easy to measure and figure out the necessary resources, like soil and fencing.

width and length

Width and length are fundamental elements when talking about rectangles, including rectangular gardens. In our problem, the width of the garden is given as 25 feet. This measurement runs from one side of the rectangle to the opposite side along the shorter distance.
The length, which we needed to find, describes the longer side from one endpoint of the width to the other. Together, these measurements—width and length—define the perimeter and area of rectangles effectively.

• Width is the measurement across a rectangle's shortest side.
• Length is the measurement across a rectangle's longest side.

Recognizing these dimensions helps in calculations like perimeter and area, which are essential when planning or constructing garden layouts.

solving for length

To find the length of a rectangular garden, one must rearrange the formula of the area of a rectangle, which is expressed as \[\text{Area} = \text{Length} \times \text{Width}.\]Solving for length involves isolating it in the formula by dividing both sides by the width:\[\text{Length} = \frac{\text{Area}}{\text{Width}}.\]In this specific problem, we know that the area is 1125 square feet and the width is 25 feet. By substituting these values into the rearranged formula, we can compute the length:\[\text{Length} = \frac{1125}{25}.\]Performing this division gives a length of 45 feet. This mathematical process is all about rearranging the formula and swapping in known values to solve for the unknown dimension.

geometry problem solving

Solving geometry problems, like finding dimensions of a rectangle, involves understanding the relationships between different geometric properties. Using formulas that relate these properties is key. In our rectangle problem, the formula for the area of a rectangle simplifies the problem-solving process.
Here are a few principles to remember:

• Identify what you know and what you need to find.
• Use known formulas and arrange them so you can find the unknown value.
• Substitute the known values into the formula.
• Execute the mathematical operations to find the missing dimension.

This structured approach is helpful across various geometry problems, encouraging logical thinking and a step-by-step method to uncover solutions.