Problem 258 - Elementary Algebra

Question

In the following exercises, solve using rectangle properties. The length of a rectangular poster is 28 inches. The area is 1316 square inches. What is the width?

Step-by-Step Solution

Answer

The width of the poster is 47 inches.

Step 1: Understand the Given Information

The length of the rectangular poster is 28 inches, and the area is 1316 square inches. We need to find the width of the poster.

Step 2: Recall the Area Formula

The area of a rectangle is calculated using the formula: \(\text{Area} = \text{Length} \times \text{Width}\).

Step 3: Set Up the Equation

Using the given information: \(\text{Area} = 28 \text{ inches} \times \text{Width}\). So, the equation is: \(\text{Width} \times 28 = 1316\).

Step 4: Solve for the Width

To find the width, divide both sides of the equation by 28: \(\text{Width} = \frac{1316}{28} \).

Step 5: Calculate the Width

Perform the division: \(\text{Width} = \frac{1316}{28} \). So, \(\text{Width} = 47 \text{ inches}\).

Key Concepts

Area of a Rectangle

In geometry, understanding how to calculate the area of a rectangle is very important. The area is a measure of how much space is enclosed within the rectangle. For a rectangle, the area can be determined using the simple formula: \(\text{Area} = \text{Length} \times \text{Width}\). This formula tells us that to get the area, we multiply the length by the width. It's important because it helps in practical scenarios like finding how much material is needed to cover a surface, like a carpet or a poster.

In our example, the length of the rectangular poster is 28 inches and the area is 1316 square inches. Using the area formula, we can set up an equation to find the unknown width by plugging in the known values.

Division

Division is one of the basic arithmetic operations and is essentially the process of determining how many times one number is contained within another. In the context of our rectangle problem, we use division to isolate the unknown variable (the width). When we set up our equation from the area formula, we have: \( \text{Width} \times 28 = 1316 \).

To solve for the width, we need to divide 1316 by 28: \( \text{Width} = \frac{1316}{28} \). This operation helps us find out how many 28-inch segments fit into 1316 square inches, essentially uncovering the length of the other dimension. Division breaks down a complex problem into simpler, more understandable parts.

Solving Equations

Solving equations is another critical mathematical skill. An equation is a mathematical statement that asserts the equality of two expressions. To solve an equation means to find the value of the variable that makes the equation true.

In this exercise, solving for the width involves following these steps:

• We start with the equation derived from the area formula: \( \text{Width} \times 28 = 1316 \).


• To isolate the width, we divide both sides by 28, resulting in the simplified equation: \( \text{Width} = \frac{1316}{28} \).


• Perform the division to find the width: \( \text{Width} = 47 \).

By methodically working through the steps, you can see that the rectangular poster must be 47 inches wide. Solving equations often involves reversing the operations that were applied to the variable, helping to uncover its value.