Question

A toy is in the form of a cone mounted on a hemisphere of radius $3.5cm.$ The total height of the toy is $15.5cm.$ Find the total surface area and volume of the toy $\left(use\pi =\frac{22}{7}\right)$.

Answer 1

Step 1: Given information

The toy in form of a cone mounted on a hemisphere.
The hemisphere's radius =$3.5cm,$ Total height of toy =$15.5cm.$

Slant height

$$\begin{gathered} =\left(l\right) \\ =\sqrt{{\left(12\right)}^2+{\left(3.5\right)}^2} \\ =\sqrt{144+12.25} \\ =\sqrt{156.25} \\ =12.5cm. \end{gathered}$$

Step 2: Required formulas

CSA of cone $=\pi rl\left(r\to radiusofconeandlistheslantheight\right)$

CSA of hemisphere$=2\pi r^2\left(r\to radiusofhemisphere\right)$

Step 3: Calculate the total surface area and the volume of the toy

We'll assume the cone is fully mounted on the hemisphere

.$$\begin{gathered} =\frac{1}{3}\times \frac{22}{7}\times \frac{35}{10}\times \frac{25}{10}\times [12+2\times \frac{35}{10}] \\ =\frac{77}{6}\times \left(12+7\right) \\ =243.83c{m}^3. \end{gathered}$$

Therefore, TSA of toy = CSA of cone + CSA of hemisphere

$$\begin{gathered} =\pi rl+2\pi r^2 \\ =\pi r\left(l+2r\right) \\ =\frac{22}{7}\times 3.5\times \left(12.5+7\right) \\ =22\times 0.5\times 19.5 \\ =214.5c{m}^2 \end{gathered}$$

Now the volume of the toy$=\frac{1}{3}\pi r^{2}h+\frac{2}{3}\pi r^3$

$$\begin{gathered} =\frac{1}{3}\times \frac{22}{7}\times \frac{35}{10}\times \frac{25}{10}\left(12+2\times \frac{35}{10}\right) \\ =\frac{77}{6}\times \left(12+7\right) \\ =243.83c{m}^3 \end{gathered}$$

Hence, the total surface area and volume of the toy is $214.5c{m}^{2}and243.83c{m}^3.$