Question

Find the ratio of the curved surface area of two cones if their diameters of the bases are equal and slant heights are in the ratio $4:3$.

Answer 1

It is given that the diameters of the bases of two cones are equal that is:

$d_1=d_2$

Divide by $2$ on both sides

$\frac{d_1}{2}=\frac{d_2}{2}.......\left(1\right)$

But the radius is half of diameter that is $\frac{d}{2}=r$, therefore, from equation 1 we get $r_1=r_2.......\left(2\right)$

It is also given that the ratio of the slant heights of the cones is $4:3$ that is:

$\frac{l_1}{l_2}=\frac{4}{3}.......\left(3\right)$

We know that the curved surface area of the cone with radius $r$ and slant height $l$ is $CSA=\pi rl$, therefore, using equations 2 and 3, we have:

$\frac{C_1}{C_2}=\frac{\pi r_1{l}_1}{\pi r_2{l}_2}\Rightarrow \frac{C_1}{C_2}=\frac{r_1}{r_1}\times \frac{4}{3}\Rightarrow \frac{C_1}{C_2}=\frac{4}{3}\Rightarrow C_1:C_2=4:3$

Hence, the ratio of the curved surface area of two cones is $4:3$.