Problem 257 - Elementary Algebra

Question

In the following exercises, solve using rectangle properties. The width of a rectangular window is 24 inches. The area is 624 square inches. What is the length?

Step-by-Step Solution

Answer

The length is 26 inches.

Step 1 - Identify the given values

The problem states that the width of the rectangular window is 24 inches and the area is 624 square inches. Let the width be denoted as \(w\) and the length as \(l\). Therefore, \(w = 24\) inches and the area \(A = 624\) square inches.

Step 2 - Write the formula for the area of a rectangle

The area of a rectangle is calculated by the formula \(A = l \times w\).

Step 3 - Substitute the given values into the formula

Substitute \(w = 24\) inches and \(A = 624\) square inches into the formula: \[ 624 = l \times 24 \ \text{or} \ 624 = 24l \]

Step 4 - Solve for \(l\)

To find the length \(l\), divide both sides of the equation by 24: \[ l = \frac{624}{24} = 26 \text{ inches} \]

Key Concepts

rectangle properties

Rectangles are everywhere in real life! They have four sides, with opposite sides being equal in length. With rectangles, there are key properties that help us solve many problems. Here are the main properties of a rectangle:
- Rectangles have four right angles (90 degrees).
- The opposite sides are both equal in length and parallel.
- The diagonals (lines connecting opposite corners) are equal in length.
These properties make rectangles easy to work with in math problems. Understanding them is important for solving any problem involving rectangles.

area of a rectangle

The area of a rectangle is how much space is inside the rectangle. To find it, we use a simple formula:
\[ A = l \times w \] Here, \(A\) stands for area, \(l\) is the length, and \(w\) is the width. The unit for area is always in square units, like square inches or square centimeters.
For example, if we have a rectangular window with a width of 24 inches and an area of 624 square inches, we can use this formula to find its length. We simply plug in the numbers:
- The width is 24 inches (\(w = 24\))
- The area is 624 square inches (\(A = 624\))
Then, we solve for the length \(l\) by manipulating the formula.

solving equations

Solving equations is vital in mathematics for finding unknown values. Here's how we solve an equation step by step using our rectangle example:
- We start with the formula for the area of a rectangle: \[ A = l \times w \]
- Next, we substitute the known values into the formula. If \(w = 24\) and \(A = 624\), then:
\[ 624 = l \times 24 \]
- To isolate \(l\), we divide both sides of the equation by 24: \[ l = \frac{624}{24} \]
- When we carry out the division, we get: \[ l = 26 \]
So, the length of the window is 26 inches. Solving equations involves isolating the variable and working step-by-step to reveal the unknown value.