Question
From a solid cylinder whose height is $16 \ cm$ and radius is $12 \ cm,$ a conical cavity of height $8 \ cm$ and of base radius $6 \ cm$ is hollowed out. Find the volume and total surface area of the remaining solid.
✓
Answer
Radius of solid cylinder $(R) = 12 \ cm$
and Height $(H) = 16 \ cm$
$\therefore \text { Volume }=\pi R ^2 H$
$=\frac{22}{7} \times 12 \times 12 \times 16$
$=\frac{50688}{7} \ cm ^3$
Radius of cone $(r) = 6 \ cm,$ and height $(h) = 8 \ cm$.
$\therefore \text { Volume }=\frac{1}{3} \pi r ^2 h$
$=\frac{1}{3} \times \frac{22}{7} \times 6 \times 6 \times 8$
$=\frac{2112}{7} \ cm ^3$
$(1)$ Volume of remaining solid
$=\frac{50688}{7}-\frac{2112}{7}$
$=\frac{48567}{7}$
$=6939.43^2 \ cm ^3$
$(2)$ Slant height of cone $I =\sqrt{ h ^2+ r ^2}$
$=\sqrt{6^2+8^2}$
$=\sqrt{36+64}$
$=\sqrt{100}$
$=10 \ cm $
Therefore, total surface area of remaining solid $=$ curved surface area of cylinder $+$ curvedsurface area of cone $+$ base area of cylinder $+$ area of circular ring on upper side of cylinder
$=2 \pi RH +\pi rl +\pi R ^2+\pi\left( R ^2- r ^2\right)$
$=\left(2 \times \frac{22}{7} \times 12 \times 16\right)+\left(\frac{22}{7} \times 6 \times 10\right)+\left(\frac{22}{7} \times 12 \times 12\right)+\left(\frac{22}{7}\left(12^2-6^2\right)\right)$
$=\frac{8448}{7}+\frac{1320}{7}+\frac{3168}{7}+\frac{22}{7}(144-36)$
$=\frac{8448}{7}+\frac{1320}{7}+\frac{3168}{7}+\frac{2376}{7}$
$=\frac{15312}{7}$
$=2187.43 \ cm ^2$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
A man standing on a cliff observes a ship at an angle of depression of the ship is 30°, approaching the shore just beneath him. Three minutes later, the angle of depression of the ship is 60°. How soon will it reach the shore?
Construct a rhombus ABCD with AB= 5.0 cm and AC= 8 cm. Draw its lines of symmetry.
The angle of elevation of a tower from a point $200 \mathrm{~m}$ from its base is $\theta$, when $\tan \theta=\frac{2}{5}$. The angle of elevation of this tower from a point $120 \mathrm{~m}$ from its base is $\Phi$. Calculate the height of tower and the value of $\Phi$.
→A( 4, 1 ), B(2,3) and C( 5,6) are the vertices of a figure which is symmetrical about x=7. Complete the figure and give the geometrical name of the figure if any.
Ramesh had Rs $100$ shares of 'Bihar Steel' paying $8\%$ dividend. He sold them at a market price of Rs $130$ and invested the proceeds in buying Rs $50$ shares of 'Jindal steel' available at Rs $75$ and paying $12\%$ dividend. He thus increased the annual income by Rs $360.$ How many shares did Ramesh sell?
→If $\tan A=n \tan B$ and $\sin A=m \sin B$, prove that:
$\cos ^2 A=\frac{m^2-1}{n^2-1}$
→$\cos ^2 A=\frac{m^2-1}{n^2-1}$
Eight metallic spheres; each of radius $2 \ mm,$ are melted and cast into a single sphere. Calculatethe radius of the new sphere.
→In the given figure, Δ ABC ~ Δ ADE. If AE: EC = 4 : 7 and DE = 6.6 cm, find BC. If 'x' be the length of the perpendicular from A to DE, find the length of perpendicular from A to BC in terms of 'x'.
Given $\left[\begin{array}{cc}2 & 1 \\ -3 & 4\end{array}\right] x=\left[\begin{array}{l}7 \\ 6\end{array}\right]$ Write the matrix $x$
→Estimate the median, the lower quartile and the upper quartile of the following frequency distribution by drawing an ogive:
| Class Interval | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 |
| Frequency | 4 | 12 | 21 | 18 | 15 | 7 | 3 |