Problem 49 - Elementary Algebra

Question

Denise wants to have a square accent of designer tiles in her new shower. She can afford to buy 625 square centimeters of the designer tiles. How long can a side of the accent be?

Step-by-Step Solution

Answer

The length of one side of the square accent is 25 centimeters.

Step 1 - Understand the Problem

Denise wants to have a square accent in her shower. She can buy 625 square centimeters of tiles. We need to find the length of one side of the square accent.

Step 2 - Recall the Formula for the Area of a Square

The area of a square is given by the formula \(\text{Area} = \text{side}^2\), where the side is the length of one side of the square.

Step 3 - Set Up the Equation

We know the area is 625 square centimeters. So, we set up the equation: \(\text{side}^2 = 625\).

Step 4 - Solve for the Side Length

To find the side length, we need to solve for \(\text{side}\). This can be done by taking the square root of both sides of the equation. \(\text{side} = \sqrt{625}\).

Step 5 - Calculate the Square Root

Calculate the square root of 625. \(\text{side} = 25\).

Step 6 - Verify the Answer

Check that the side length squared gives the area. \((25)^2 = 625\), so the side length is correct.

Key Concepts

Area of a Square

The area of a square is a measure of the space contained within its boundaries. It is calculated by squaring the length of one of its sides. In mathematical terms:
\[\text{Area} = \text{side}^2\]
For example, if each side of a square is 5 cm, then its area is:
\[\text{Area} = 5^2 = 25 \text{ cm}^2\]
In Denise's case, she needs to cover an area of 625 square centimeters with tiles. By understanding this, we can solve for the side length of the square.

Equation Solving

Solving equations involves finding unknown values that make the equation true. In our problem, we know the area (625 cm²) and need to find one side length of the square. We set up our equation using the formula for the area of a square:
\[\text{side}^2 = 625\]
To solve for the side, we need to isolate the variable. Here, the variable is 'side'. Take the square root of both sides of the equation:
\[\text{side} = \sqrt{625}\]
This operation unlocks the value we need, guiding us to the solution.

Square Root Calculation

The square root of a number is a value that, when multiplied by itself, gives the original number. To find the side length of Denise's square accent, we need to calculate \(\text{side} = \sqrt{625}\). Here's how you can do it:
Identify perfect squares, which are numbers like 1, 4, 9, 16, and so on. Recognize that 625 is one of these perfect squares.
Calculate the square root:
\[\text{side} = 25\]
This means that each side of the square accent should be 25 cm.

Geometry Basics

Geometry is the branch of mathematics dealing with shapes, sizes, and properties of space. One of the simplest 2D shapes in geometry is the square.
A square has four equal sides and four right angles. It is a regular polygon. When dealing with squares, concepts such as perimeter (sum of all sides) and area are fundamental.
For instance, given a side length (s), the perimeter (P) of a square is:
\[\text{P} = 4s\]
And, as we used earlier, the area (A) is:
\[\text{A} = s^2\]
Using these basics helps tackle various problems involving squares, like Denise's designer tile problem.