Question

The base of a conical tent is of area $616\mathrm{sq}$. $\mathrm{cm}$. A $48\mathrm{cm}$ long vertical pole is placed at its centre so that it touches the roof of the tent.

How much canvas is needed to make the tent if the base is also covered with canvas?

• A. $2816{\mathrm{cm}}^2$
• B. $2861{\mathrm{cm}}^2$
• C. $2618{\mathrm{cm}}^2$
• D. $2681{\mathrm{cm}}^2$

Answer 1

The correct option is A

$2816{\mathrm{cm}}^2$

Step 1: Given data

Area of base of conical tent,${\mathrm{\pi r}}^2=616\mathrm{sq}.\mathrm{cm}$

Height of conical tent, $h=48cm$

Amount of canvas required to make the conical tent$=?$

Total surface are of cone$=\mathrm{\pi r}(\mathrm{r}+\mathrm{l})$

$l=\sqrt{r^2+h^2}$

Step 2: Calculating the amount of canvas required

$$\begin{gathered} {\mathrm{\pi r}}^2=616 \\ \frac{22}{7}\times r^2=616 \\ {r}^2=\frac{616\times 7}{22}=196 \\ r=14cm \end{gathered}$$

Calculating the slant height, $l$ of the conical tent-

$$\begin{gathered} l=\sqrt{r^2+h^2} \\ l=\sqrt{{14}^2+{48}^2}=\sqrt{196+2304}=\sqrt{2500} \\ l=50cm \end{gathered}$$

We know that, the amount of canvas required to make the conical tent will be equal to its total surface area.

Thus, Total surface area of conical tent is given as,

$$\begin{gathered} \mathrm{\pi r}(\mathrm{r}+\mathrm{l})=\frac{22}{7}\times 14(14+50) \\ =22\times 2\left(64\right)=2816c{m}^2 \end{gathered}$$

Hence, option A is correct.