Question : The volume of a cone with a height equal to the radius and slant height of 5 cm is:
Option 1: $\frac{125 \pi}{12 \sqrt{3}} \mathrm{~cm}^3$
Option 2: $\frac{125 \pi}{6 \sqrt{3}} \mathrm{~cm}^3$
Option 3: $\frac{125 \pi}{12 \sqrt{2}} \mathrm{~cm}^3$
Option 4: $\frac{125 \pi}{6 \sqrt{2}} \mathrm{~cm}^3$
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Correct Answer: $\frac{125 \pi}{6 \sqrt{2}} \mathrm{~cm}^3$
Solution : Given: A cone with a height equal to the radius and a slant height of 5 cm. Using Pythagoras theorem, we have, ⇒ $r^{2}+r^{2}=5^{2}$ ⇒ $2r^{2}=25$ ⇒ $r=\frac{5}{\sqrt{2}}$ Volume of the cone = $\frac{1}{3}\pi r^{2}h$ ---------(where $r=h$) Putting the values, we have, ⇒ $\frac{1}{3}\pi ×\frac{5}{\sqrt{2}}×\frac{5}{\sqrt{2}}×\frac{5}{\sqrt{2}}$ ⇒ $\frac{125}{6\sqrt{2}}\pi \mathrm{~cm}^3$ Hence, the correct answer is $\frac{125}{6\sqrt{2}}\pi \mathrm{~cm}^3$.
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Question : The ratio of the height and the diameter of a right circular cone is 6: 5 and its volume is $\frac{2200}{7} \mathrm{~cm}^3$. What is its slant height? (Take $\pi=\frac{22}{7}$ )
Question : If the curved surface area of a cylinder is $126\pi$ cm2 and its height is 14 cm, what is the volume of the cylinder?
Question : The slant height and radius of a right circular cone are in the ratio 29 : 20. If its volume is $4838.4 \pi ~\mathrm{cm}^3$, then its radius is:
Question : The area of a sector of a circle is 88 sq. cm., and the angle of the sector is 45°. Find the radius of the circle. (Use $\pi=\frac{22}{7}$)
Question : In a circle, a diameter AB and a chord PQ (which is not a diameter) intersect each other at X perpendicularly. If AX : BX = 3 : 2 and the radius of the circle is 5 cm, then the length of the chord PQ is:
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