Calculating the Dimensions of a Rectangle Given Its Area

Understanding how to find the dimensions of a rectangle using its area is a fundamental concept in algebra and geometry. This guide will walk you through a step-by-step process to solve such problems using a real-world example. Specifically, we will find the length and width of a rectangle with an area of 24 square centimeters given that the width is 2 cm less than the length.

Example Problem

Given that the area of a rectangle is 24 square centimeters and the width is 2 cm less than the length, what are the length and width of the rectangle?

Step-by-Step Solution

Define Variables

To solve this, we first need to define our variables.

Length - Let the length of the rectangle be (L) cm. Width - Since the width is 2 cm less than the length, we express the width as (W L - 2).

Set Up the Equation

The area of a rectangle is given by the formula:

[text{Area} L times W]

Substituting the expression for width into the area formula, we get:

[L times (L - 2) 24]

Expand and Rearrange

Expanding the equation, we obtain:

[L^2 - 2L 24]

Rearranging it into standard quadratic form, we get:

[L^2 - 2L - 24 0]

Solve the Quadratic Equation

Using the quadratic formula to solve for (L):

[L frac{-b pm sqrt{b^2 - 4ac}}{2a}]

Here, (a 1), (b -2), and (c -24). Substituting these values in, we get:

[L frac{-(-2) pm sqrt{(-2)^2 - 4 cdot 1 cdot (-24)}}{2 cdot 1}]

[L frac{2 pm sqrt{4 96}}{2}]

[L frac{2 pm sqrt{100}}{2}]

[L frac{2 pm 10}{2}]

This gives us two potential solutions:

(L frac{12}{2} 6) or (L frac{-8}{2} -4))

Since the length cannot be negative, we take (L 6).

Find the Width

Substituting (L 6)) back to find (W)):

(W L - 2 6 - 2 4) cm.

Conclusion

The length of the rectangle is 6 cm and the width is 4 cm.

Thus, the dimensions of the rectangle are:

Length: 6 cm Width: 4 cm

Additional Examples

For a more concrete understanding, let's consider a similar problem:

Create a right triangle with an area of 24 sq cm where one leg is 2 cm less than the other. Using the area formula (A frac{1}{2} cdot a cdot b), we get:

(24 frac{1}{2} cdot x cdot (x - 2))

Solving this, we find the legs to be 6 cm and 4 cm.

Real-World Application

Understanding these calculations can be useful in various fields, from construction and architecture to product design and manufacturing.

Conclusion

By breaking down the problem into simple steps and using algebraic methods, we can easily find the dimensions of a rectangle given its area and a relationship between its dimensions.