Problem 256 - Elementary Algebra

Question

In the following exercises, solve using rectangle properties. The area of a rectangle is 782 square centimeters. The width is 17 centimeters. What is the length?

Step-by-Step Solution

Answer

The length of the rectangle is 46 centimeters.

Step 1: Understand the Problem

Given the area of a rectangle is 782 square centimeters and the width is 17 centimeters. We need to find the length of the rectangle.

Step 2: Write the Area Formula

The area of a rectangle is given by the formula: \[ \text{Area} = \text{Length} \times \text{Width} \]

Step 3: Plug in the Known Values

Substitute the given values into the formula. We have the area (782 cm²) and the width (17 cm): \[ 782 = \text{Length} \times 17 \]

Step 4: Solve for Length

To find the length, divide both sides of the equation by 17: \[ \text{Length} = \frac{782}{17} \] Calculate the quotient: \[ \text{Length} = 46 \]

Step 5: Conclude the Solution

The length of the rectangle is 46 centimeters.

Key Concepts

area of a rectangle

The area of a rectangle is a basic but essential concept in geometry and algebra. To understand it fully, you need to know that a rectangle is a four-sided figure with opposite sides that are equal in length. The area of a rectangle is the amount of space enclosed within its four sides.

Knowing how to calculate the area is crucial because it helps in various real-life scenarios, such as determining the amount of paint needed to cover a wall or the size of carpet required to cover a floor.

The formula to find the area of a rectangle is:
\[\text{Area} = \text{Length} \times \text{Width}\]

In our exercise, we are given the area (782 cm²) and the width (17 cm). By understanding this formula, we can rearrange it to find the missing length.

length calculation

Calculating length is another important aspect in algebra and geometry. Often, we know one dimension and the area but need to figure out the missing dimension. This is where algebra comes into play.

To find the length, you need to manipulate the area formula of a rectangle. Here is how you do it step-by-step:

• Start with the area formula: \[\text{Area} = \text{Length} \times \text{Width}\]
• Substitute the known values into the equation. For our problem: \[782 = \text{Length} \times 17\]
• To isolate length, divide both sides by the width: \[\text{Length} = \frac{782}{17}\]
• Perform the division to find the length: \[\text{Length} = 46\]

Thus, the length of the rectangle is 46 centimeters. By practicing these steps, you can master the technique of calculating any missing dimension when given the area and one other dimension.

basic algebra

Basic algebra is fundamental for solving a wide range of mathematical problems, including geometry problems like the one in our exercise.

Algebra involves using symbols and letters to represent numbers and quantities in equations and formulas. These symbols and equations help us to solve for unknown values.

In our exercise, we're using basic algebra to rearrange the area formula and solve for the missing length. Here are the key steps:

• Start with the given formula: \[\text{Area} = \text{Length} \times \text{Width}\]
• Substitute the known values to form an equation: \[782 = 17 \times \text{Length}\]
• Isolate the unknown variable (Length): Divide both sides by the known value (Width), so it's: \[\text{Length} = \frac{782}{17}\]
• Simplify the division to solve for the length: \[\text{Length} = 46\]

Understanding basic algebraic manipulations such as these allows you to solve similar problems effectively. Remember, algebra is just a tool – like learning to read or write – that helps you unlock solutions to various problems.