Question

Sum of radii of two circles is $140\mathrm{cm}$ and the difference of their circumferences is $88\mathrm{cm}$. Find the diameters of the circles.

Answer 1

Step 1: Determine the radii of two circles.

Let $r_1,r_2$ are radius of the circles

The sum of radii of two circles $\left(r_1+r_2\right)$ is $140\mathrm{cm}$ and the difference of their circumferences $\left(C_1-C_2\right)$ is $88\mathrm{cm}$.

The circumference of a circle $C$ is calculated by $C=2\mathrm{\pi r}$ where $r$ represents the radius of a circle.

$$\begin{gathered} C_1-C_2=88 \\ \Rightarrow 2{\mathrm{\pi r}}_1-2{\mathrm{\pi r}}_2=88 \\ \Rightarrow {\mathrm{r}}_1-{\mathrm{r}}_2=\frac{88}{2\mathrm{\pi }} \\ \Rightarrow {\mathrm{r}}_1-{\mathrm{r}}_2=\frac{88\times 7}{2\times 22} \\ \Rightarrow {\mathrm{r}}_1=14+r_2 \end{gathered}$$

Substitute $14+r_2$ for $r_1$ in $r_1+r_2=140$.

$$\begin{gathered} 14+r_2+r_2=140 \\ \Rightarrow 2r_2=126 \\ \Rightarrow r_2=63\mathrm{cm} \end{gathered}$$

Substitute $63$ for $r_2$ in ${\mathrm{r}}_1=14+r_2$.

$\begin{array}{rcl}{\mathrm{r}}_1& =& 14+63\\ & =& 77\mathrm{cm}\end{array}$

Step 2: Determine the diameter of the two circles

The diameter of a circle $\left(d\right)$ is calculated by $d=2r$.

The diameter of a circle $\left(d_1\right)$ is calculated by $d_1=2\times 77=154cm$.

The diameter of a circle $\left(d_2\right)$ is calculated by $d_2=2\times 63=126cm$.

Hence, the diameter of the two circles are $154\mathrm{cm}$ and $126\mathrm{cm}$ respectively.