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Question Detail
The radii of two cones are in ratio 2:1, their volumes are equal. Find the ratio of their heights.
- 1:4
- 1:3
- 1:2
- 1:5
Answer: Option A
Explanation:
Let their radii be 2x, x and their heights be h and H resp.
Then,
\begin{aligned}
\text{Volume of cone =}\frac{1}{3}\pi r^2h \\
\frac{\frac{1}{3}*\pi *{(2x)}^2*h}{\frac{1}{3}*\pi *{x}^2*H} \\
=> \frac{h}{H} = \frac{1}{4} \\
=> h:H = 1:4
\end{aligned}
1. The radii of two cones are in ratio 2:1, their volumes are equal. Find the ratio of their heights.
- 1:4
- 1:3
- 1:2
- 1:5
Answer: Option A
Explanation:
Let their radii be 2x, x and their heights be h and H resp.
Then,
\begin{aligned}
\text{Volume of cone =}\frac{1}{3}\pi r^2h \\
\frac{\frac{1}{3}*\pi *{(2x)}^2*h}{\frac{1}{3}*\pi *{x}^2*H} \\
=> \frac{h}{H} = \frac{1}{4} \\
=> h:H = 1:4
\end{aligned}
2. If the volume of two cubes are in the ratio 27:1, the ratio of their edges is:
- 3:1
- 3:2
- 3:5
- 3:7
Answer: Option A
Explanation:
Let the edges be a and b of two cubes, then
\begin{aligned}
\frac{a^3}{b^3} = \frac{27}{1} \\
=> \left( \frac{a}{b} \right)^3 = \left( \frac{3}{1} \right)^3 \\
\frac{a}{b}=\frac{3}{1} \\
=> a:b = 3:1
\end{aligned}
3. 66 cubic centimetres of silver is drawn into a wire 1 mm in diameter. The length if the wire in meters will be:
- 76 m
- 80 m
- 84 m
- 88 m
Answer: Option C
Explanation:
Let the length of the wire be h
\begin{aligned}
Radius = \frac{1}{2}mm = \frac{1}{20}cm\\
\pi r^2h = 66 \\
\frac{22}{7}*\frac{1}{20}*\frac{1}{20}*h = 66 \\
=> h = \frac{66*20*20*7}{22} \\
= 8400 cm \\
= 84 m
\end{aligned}
4. Two right circular cylinders of equal volumes have their heights in the ratio 1:2. Find the ratio of their radii.
- \begin{aligned} \sqrt{3}:1 \end{aligned}
- \begin{aligned} \sqrt{7}:1 \end{aligned}
- \begin{aligned} \sqrt{2}:1 \end{aligned}
- \begin{aligned} 2:1 \end{aligned}
Answer: Option C
Explanation:
Let their heights be h and 2h and radii be r and R respectively then.
\begin{aligned}
\pi r^2h = \pi R^2(2h) \\
=> \frac{r^2}{R^2} = \frac{2h}{h} \\
= \frac{2}{1} \\
=> \frac{r}{R} = \frac{\sqrt{2}}{1} \\
=> r:R = \sqrt{2}:1 \\
\end{aligned}
5. A circular well with a diameter of 2 meters, is dug to a depth of 14 meters. What is the volume of the earth dug out.
- \begin{aligned} 40 m^3 \end{aligned}
- \begin{aligned} 42 m^3 \end{aligned}
- \begin{aligned} 44 m^3 \end{aligned}
- \begin{aligned} 46 m^3 \end{aligned}
Answer: Option C
Explanation:
\begin{aligned}
Volume = \pi r^2h \\
Volume = \left(\frac{22}{7}*1*1*14\right)m^3 \\
= 44 m^3
\end{aligned}
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