Question

this series of questions will build your fluency with interpreting scaling relationships in area calculations. for each question below, analyze the structure of the area calculation to initially answer the questions without computation. recall that the area of a circle is

$a = \pi r^{2}$, where $r$ is the radius.

• use the answer key to determine that you accurately answered each question.
• use computation to confirm that your solution is accurate.
• if necessary, revise your work.

1. by what factor will the area change when:

a. a circle with an area of $10\\ cm^{2}$ has its radius doubled in size?

b. a triangle with an area of $6\\ cm^{2}$ is scaled down to a fifth of its size?

1. did knowing the initial area affect your answers to question 1? explain.

Explanation:

Step1: Analyze the area formula for a circle

The area formula for a circle is \(A = \pi r^{2}\). If the radius \(r\) is doubled (\(r_{new}=2r\)), then the new area \(A_{new}=\pi(2r)^{2}\).

Step2: Simplify the new - area formula

Using the formula \((ab)^{n}=a^{n}b^{n}\), we have \(A_{new}=\pi\times4r^{2}=4(\pi r^{2})\). Since \(A = \pi r^{2}\), the factor by which the area changes is \(4\).

Step3: Analyze the scaling of a triangle's area

For any two - dimensional shape, if the linear scale factor is \(k\), the area scale factor is \(k^{2}\). When a triangle is scaled down to \(\frac{1}{5}\) of its size (linear scale factor \(k = \frac{1}{5}\)), the area scale factor is \(k^{2}\).

Step4: Calculate the area scale factor for the triangle

Using the formula for the area scale factor of a two - dimensional shape (\(A_{scale}=k^{2}\)), with \(k=\frac{1}{5}\), we get \(A_{scale}=(\frac{1}{5})^{2}=\frac{1}{25}\).

Answer:

a. The area changes by a factor of \(4\).
b. The area changes by a factor of \(\frac{1}{25}\).