Problem 28

Question

A square of side $x$ has area $x^{2} .$ By what factor does the area change if the length is (a) Doubled? (b) Tripled? (c) Halved? (d) Multiplied by $0.1 ?$

Step-by-Step Solution

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Answer

Answer: (a) The area increases by a factor of 4. (b) The area increases by a factor of 9. (c) The area decreases by a factor of 1/4. (d) The area decreases by a factor of 0.01.

1Step 1: (a) Doubling side length

(If the side length is doubled, the new side would be $2x$. Calculate the area of the square with this new side length and find the ratio of the new area to the old area.) New side length = $2x$ New area = $(2x)^2 = 4x^2$ Factor by which the area changes = $\frac{4x^2}{x^2} = 4$ So, when the side length is doubled, the area is increased by a factor of 4.

2Step 2: (b) Tripling side length

(If the side length is tripled, the new side would be $3x$. Calculate the area of the square with this new side length and find the ratio of the new area to the old area.) New side length = $3x$ New area = $(3x)^2 = 9x^2$ Factor by which the area changes = $\frac{9x^2}{x^2} = 9$ So, when the side length is tripled, the area is increased by a factor of 9.

3Step 3: (c) Halving side length

(If the side length is halved, the new side would be $\frac{1}{2}x$. Calculate the area of the square with this new side length and find the ratio of the new area to the old area.) New side length = $\frac{1}{2}x$ New area = $(\frac{1}{2}x)^2 = \frac{1}{4}x^2$ Factor by which the area changes = $\frac{\frac{1}{4}x^2}{x^2} = \frac{1}{4}$ So, when the side length is halved, the area is decreased by a factor of $\frac{1}{4}$.

4Step 4: (d) Multiplying side length by 0.1

(If the side length is multiplied by 0.1, the new side would be $0.1x$. Calculate the area of the square with this new side length and find the ratio of the new area to the old area.) New side length = $0.1x$ New area = $(0.1x)^2 = 0.01x^2$ Factor by which the area changes = $\frac{0.01x^2}{x^2} = 0.01$ So, when the side length is multiplied by 0.1, the area is decreased by a factor of 0.01.

Key Concepts

Area of a squareScaling factorsSide length transformations

Area of a square

Understanding the area of a square is fundamental in geometry. The area measures how much space is contained within the square. If a square has side length denoted by $x$, its area is simply calculated by squaring this side length. Hence, the formula for the area of a square is:

$ \text{Area} = x^2 $

This expression represents the total number of square units within the square. Calculating the area requires multiplying the length of one side by itself.
For instance, if a square's side measures 3 units, its area would be $3 \times 3 = 9$ square units. Simple and straightforward, right?
Knowing how to work out the area of a square is essential because it sets the stage for understanding how changes in side length impact the area as covered next.

Scaling factors

Scaling factors help us understand how different transformations affect the size of a shape. When we increase or decrease the side length of a square, the scaling factor determines how much larger or smaller the area becomes.
For example:

If we double the side length from $x$ to $2x$, the new area becomes $(2x)^2 = 4x^2$. The area increases by a factor of 4 due to the square of the scaling factor: $2^2 = 4$.
If we triple the side length to $3x$, then $(3x)^2 = 9x^2$. The area scales by a factor of 9, as $3^2 = 9$.

Understanding scaling factors is vital. It helps us see how a linear change in dimensions can have a squared impact on area, which is not always intuitive. By grasping this concept, we can predict how transformations affect shapes in geometry, making it a powerful tool in both simple and complex calculations.

Side length transformations

Changes to the side length of a square can either increase or decrease its area dramatically. Let’s explore side length transformations and how they affect your square.
When altering a side's length, remember:

If the side is halved, the new side length is $\frac{1}{2}x$. The area becomes $(\frac{1}{2}x)^2 = \frac{1}{4}x^2$. This reduces the area by a factor of $\frac{1}{4}$.
On the other hand, multiplying the side by 0.1 ($0.1x$) results in an area of $(0.1x)^2 = 0.01x^2$. The area shrinks significantly, by a factor of 0.01.

These transformations demonstrate the dramatic effects that small changes to dimensions can have on the overall area. Side length transformations teach us to consider not just how much a side length changes, but also that any percentage change in length results in a pronounced percentage change in area, squared in nature. This understanding is crucial for precise calculations in real-world geometry applications.

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Problem 28

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