Question

the figures below are similar.

two circles, one pink with radius 5 m, one blue with radius u

if the ratio of the circles’ circumferences is 5:3, what is u?

u = \boxed{} meters

Explanation:

Step1: Recall the formula for circumference

The formula for the circumference of a circle is \(C = 2\pi r\). For two similar circles (all circles are similar), if the radii are \(r_1\) and \(r_2\), the ratio of their circumferences \(C_1:C_2=(2\pi r_1):(2\pi r_2)=r_1:r_2\).

Step2: Set up the proportion

Let \(r_1 = 5\) (radius of the first - circle) and \(r_2=u\) (radius of the second - circle). Given that \(\frac{C_1}{C_2}=\frac{5}{3}\), and since \(\frac{C_1}{C_2}=\frac{r_1}{r_2}\), we have the proportion \(\frac{5}{u}=\frac{5}{3}\).

Step3: Solve the proportion for \(u\)

Cross - multiply: \(5\times3 = 5\times u\). Then \(15 = 5u\). Divide both sides by 5: \(u=\frac{15}{5}=3\).

Answer:

\(3\)