Question

from a solid cylinder of height 30 cm and radius 7 cm a conical cavity of height 24cm and base radius 7 cm is drilled out.Find Volume and the Total Surface area of remaining solid

Answer 1

Solution-

1. Volume

$V_{\text{remaining solid}}=V_{\text{cylinder}}-V_{\text{cone}}$

$V_{\text{remaining solid}}=[\pi\times (\text{base radius})^2\times \text{height}}]-[\pi\times(\text{base radius})^2\times \frac{\text{height}}{3}]$

Putting the values,

$\Rightarrow V_{\text{remaining solid}}=(\pi\times 7^2\times 30)-(\pi\times 7^2\times\frac{24}{3})$

$\Rightarrow V_{\text{remaining solid}}=(\pi\times 49\times 30)-(\pi\times 49\times 8)$

$\Rightarrow V_{\text{remaining solid}}=\pi\times 49\times 22$

$\Rightarrow V_{\text{remaining solid}}=3386.6\ cm^3$

2. Surface Area

$A_{\text{remaining solid}}=A_{\text{curved area of cylinder}}+A_{\text{area of base circle}}+A_{\text{curved area of cone}}$

$A_{\text{remaining solid}}=(2\pi\times \text{base radius}\times \text{height})+(\pi \times (\text{base radius})^2)+(\pi\times \text{base radius}\times \text{slant height})$

Putting the values,

$\Rightarrow A_{\text{remaining solid}}=(2\pi\times 7\times 24)+(\pi\times 7^2)+(\pi \times 7\times \sqrt{24^2+7^2})$

$\Rightarrow A_{\text{remaining solid}}=(2\pi\times 7\times 24)+(\pi\times 49)+(\pi \times 7\times 25)$

$\Rightarrow A_{\text{remaining solid}}=336\pi+49\pi+175\pi$

$\Rightarrow A_{\text{remaining solid}}=560\pi$

$\Rightarrow A_{\text{remaining solid}}=1759.3\ cm^2$