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Chapter 11: Problem 108

The area of a square room is 225 square feet. Find the dimensions of the room.

Short Answer

Expert verified
The room's dimensions are 15 feet by 15 feet.

Step by step solution

01

Identify the Formula for Area of a Square

The area of a square is calculated using the formula \( A = s^2 \), where \( A \) is the area and \( s \) is the side length of the square. In this problem, we are given that \( A = 225 \) square feet.
02

Solve for the Side Length

To find the side length \( s \) of the square, we need to solve the equation \( s^2 = 225 \). This can be done by taking the square root of both sides: \( s = \sqrt{225} \).
03

Calculate the Square Root

Calculate \( \sqrt{225} \). Since \( 15 \times 15 = 225 \), it follows that \( \sqrt{225} = 15 \).
04

Conclude the Dimensions

Since the side length \( s \) is 15 feet, the dimensions of the square room, which are the lengths of its sides, are 15 feet by 15 feet.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Area of a Square
The area of a square is a simple geometric concept that helps us understand the size of a two-dimensional space within a square. To find the area of a square, you multiply the side length by itself, or raise it to the power of two.
This relationship is mathematically expressed as:
  • \( A = s^2 \)
where \( A \) is the area and \( s \) is the length of one side of the square.
Squares have equal side lengths, making them unique and easy to calculate.
In practical situations, knowing the area is important for planning and determining space use, such as in floor plan design or landscaping.
Square Root Calculation
Calculating the square root is the process of finding a number that, when multiplied by itself, yields the original number.
For example, in the equation \( s^2 = 225 \), we solve for \( s \) by finding \( \sqrt{225} \).
Here, since \( 15 \times 15 = 225 \), we determine \( \sqrt{225} = 15 \).
Square roots are often found using mental math, estimation, or a calculator. They are useful for finding dimensions from areas or breaking down larger calculations.
  • When dealing with perfect squares like 225, you can often find the root easily.
  • If the number isn't a perfect square, you may need a calculator for a precise value.
Algebraic Equation Solving
Solving algebraic equations is a fundamental skill in mathematics, which involves finding the variable's value that makes the equation true. In our scenario, the equation \( s^2 = 225 \) is a basic algebraic equation.
To solve for \( s \), you perform the operation that reverses the power of two, which is taking the square root on both sides of the equation. This yields \( s = \sqrt{225} = 15 \).
A systematic approach to solving equations includes:
  • Understanding the equation and identifying what needs solving.
  • Applying algebraic rules to isolate the variable.
  • Performing operations equally on both sides to maintain equality.
This ensures the integrity of the equation and helps find the correct solution efficiently.

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Most popular questions from this chapter

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. $$ f(x)=-(x-2)^{2} $$

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. $$ f(x)=-3(x+2)^{2}+2 $$

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. $$ g(x)=-\frac{3}{2}(x-1)^{2}-5 $$

Add the proper constant to each binomial so that the resulting trinomial is a perfect square trinomial. See Section 11.1. $$ z^{2}-3 z $$

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. $$ H(x)=\frac{3}{4} x^{2}-2 $$
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