Question

The radius and slant height of a cone are in the ratio \[3:5\]. If its curved surface area is 2310 cm\[^2\], find its radius.

Answer 1

Hint: Here, we will use ratio of radius and slant height to find the value of slant height \[l\] in terms of radius \[r\]. Then we will use the curved surface area of the cone \[\pi rl\] to find the radius.

Complete step-by-step solution
Given that the curved surface area is 2310 cm\[^2\].

Let the radius of a cone is \[r\] and slant height of a cone is \[l\].

It is given that the ratio of the radius and slant height of a cone is \[3:5\].

Rewriting the equation \[\dfrac{r}{l} = \dfrac{3}{5}\] in terms of \[r\], we get
\[
   \Rightarrow 5r = 3l \\
   \Rightarrow l = \dfrac{5}{3}r \\
\]

Substituting this value of \[r\] in the formula of the curved surface area of cone \[\pi rl\], we get

\[{\text{Curved surface area}} = \pi r\left( {\dfrac{5}{3}r} \right)\]

Substituting the value of curved surface area in the above equation, we get

\[ \Rightarrow 2310 = \dfrac{5}{3}\pi {r^2}\]
Multiplying the above equation by 3 on each of the sides, we get
\[ \Rightarrow 6930 = 5\pi {r^2}\]

Dividing the above equation by 5 on both sides, we get

\[
   \Rightarrow \dfrac{{6930}}{5} = \pi {r^2} \\
   \Rightarrow 1386 = \pi {r^2} \\
\]

Using the value of \[\pi \] in the above equation, we get

\[ \Rightarrow \dfrac{{22}}{7}{r^2} = 1596\]

Multiplying the above equation by \[\dfrac{7}{{22}}\] on both sides, we get

\[
   \Rightarrow \dfrac{7}{{22}} \times {r^2} = \dfrac{7}{{22}} \times 1386 \\
   \Rightarrow {r^2} = 7 \times 63 \\
   \Rightarrow {r^2} = 441 \\
\]

Taking square root of the above equation, we get

\[
   \Rightarrow r = \pm \sqrt {441} \\
   \Rightarrow r = \pm 21{\text{ cm}} \\
\]

Since the radius can never be negative, the negative value is discarded.

Thus, the radius is 21 cm.

Note: Some students use the formula of the total surface area instead of the curved surface area of a cone. The curved surface area is defined as the area of only curved surfaces leaving the circular top and base. Whereas the total surface area is the area of the curved surface area along with bases. This is the main difference between the CSA and TSA, students should always keep this in mind.