Problem 62 - Elementary Algebra

Question

If the sides of a square are doubled, then by what factor is the area increased? Why?

Step-by-Step Solution

Answer

The area increases by a factor of 4 because \( (2s)^2 = 4s^2 \).

Step 1: Understand the Formula for Area of a Square

The area of a square is calculated using the formula \( A = s^2 \), where \( s \) is the length of a side of the square.

Step 2: Calculate the Original Area

Assume the original side length is \( s \). The original area is \( A_{original} = s^2 \).

Step 3: Consider the New Side Length

When the side length is doubled, the new side length is \( 2s \).

Step 4: Calculate the New Area

The new area with the side length doubled is \( A_{new} = (2s)^2 = 4s^2 \).

Step 5: Determine the Factor of Increase

To find by what factor the area increased, divide the new area by the original area: \( \frac{A_{new}}{A_{original}} = \frac{4s^2}{s^2} = 4 \).

Key Concepts

Geometry and the Area of a Square

In geometry, understanding the concept of a square is fundamental. A square is a special type of rectangle where all four sides are of equal length. To find the area of a square, we use the formula \( A = s^2 \), where \( s \) represents the side length of the square.
The area measures the space contained within the boundaries of the square.
This concept is important because it helps us understand how different changes to the square can affect its area.

• If you increase the side length, the area increases.
• If you reduce the side length, the area decreases.

By understanding the relationship between side length and area, we can solve many practical problems involving spaces, surfaces, and physical dimensions.

Doubling Side Length

When we talk about doubling the side length of a square, we mean multiplying the side length by 2. If the original side length is \( s \), then the new side length becomes \( 2s \).
This change significantly affects the area of the square.
Let's see why this happens:

• The original area is determined by \( s^2 \).
• When the side length doubles, the new square's area is calculated by \( (2s)^2 \).

Upon expanding \( (2s)^2 \), we see it equals \( 4s^2 \).
This tells us that doubling the side length does not simply double the area; it quadruples it.
Understanding this demonstrates how changes in dimensions have exponential effects on area.

Area Increase Factor

The concept of an area increase factor tells us how much the area grows when we change the dimensions of a shape.
For the case of a square with a doubled side length, the increase factor is quite clear.
To find the increase factor, we calculate the ratio of the new area to the original area:

• Original area : \( s^2 \)
• New area : \( 4s^2 \)

Dividing the new area by the original area gives us \( \frac{4s^2}{s^2} = 4 \).
This indicates that the area increases by a factor of 4 when the side length is doubled.
This concept is crucial in geometry as it helps predict how alterations in size affect overall area. It's a practical way to understand the geometric principles at play.