Question

The ratio of circumference of two circles is $3:2$. The radius of the smaller circle is 8 inches. What is the radius of the larger circle?

Answer 1

Hint:We first try to form the proportionality relation between the radius and the circumference of the circle. We take an arbitrary constant. We use the given values of the variables to find the value of the constant. Finally, we put the constant’s value to find the radius of the larger circle.

Complete step by step answer:
If the radius of a circle is r, then the circumference is $2\pi r$. The two terms are directly proportional. We have been given the relation between two variables where we assume the radius as r and ratio number for circumference as t.The relation between r and t is a direct relation. It’s given r varies directly as t which gives $r\propto t$.

To get rid of the proportionality we use the proportionality constant which gives $r=kt$.Here, the number k is the proportionality constant.It’s given $r=8$ when $t=2$.We put the values in the equation $r=kt$to find the value of k. So, $8=k\times 2$. Simplifying we get
\[8=k\times 2 \\
\Rightarrow k=\dfrac{8}{2}=4 \\ \]
Therefore, the equation becomes with the value of k as $r=4t$.Now we simplify the equation to get the value of r for the ratio number for circumference being 3.
\[r=4t \\
\therefore r=4\times 3=12 \\ \]
Therefore, the radius of the larger circle is 12 inches.

Note: In a direct proportion, the ratio between matching quantities stays the same if they are divided. We can also use the $2\pi r$ to find circumference for smaller circles and use the ratio to find the circumference for smaller circles. That gives the radius of the larger circle