Question

(a)A circle has a radius of 12cm. If the radius is increased by a factor of 7, how's many times larger will the circle's circumference be? (b)What is the scale factor of the circumference of a circle with a radius of 3 when compared to the original circle? Write your answer as a decimal to the nearest hundredth.

Answer 1

(a)

The circumference of a circle is found using the formula,

`C=2\pi r`

Where

C is the circumference

r is the radius

A circle has radius 12 cm, so its circumference is

`\begin{gathered} C_1=2\pi r \\ C_1=2\pi(12) \\ C_1=24\pi \end{gathered}`

When the radius (12 cm) is increased by a factor of 7, we mean the radius is 7 times the original, so, the new radius is

`7*12=84`

So, the circumference of this circle will be >>>>

`\begin{gathered} C_2=2\pi r \\ C_2=2\pi(84) \\ C_2=168\pi \end{gathered}`

To find how many times larger this circumference is of the previous one, we divide,

`(C_2)/(C_1)=(168\pi)/(24\pi)=7_{}`

Answer7

(b)

Using the circumference formula, we can find the circumference of this circle,

`\begin{gathered} C=2\pi r \\ C=2\pi(3) \\ C=6\pi \end{gathered}`

The original circle had a circumference of 24π

The question basically asks "how many times is the circumference of this circle compared to the circumference of the original circle?"

Of couse, it will be less than 1 since this circle is smaller than the original one.

So,

`\begin{gathered} (6\pi)/(24\pi) \\ =(1)/(4) \end{gathered}`

So, the scale factor is 1/4, or 0.25

Answer0.25