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Question Detail
The surface area of a sphere is same as the curved surface area of a right circular cylinder whose height and diameter are 12 cm each. The radius of the sphere is:
- 4 cm
- 6 cm
- 8 cm
- 10 cm
Answer: Option B
Explanation:
\begin{aligned}
\text{Curved surface area of sphere =}\\
\frac{4}\pi r^2 \\
\text{Surface area of cylinder =} \\
2\pi rh \\
=> \frac{4}\pi r^2 = 2\pi rh \\
=> r^2 = \frac{6*12}{2} \\
=> r^2 = 36 \\
=> r = 6
\end{aligned}
Note: Diameter of cylinder is 12 so radius is taken as 6.
1. A cone of height 9 cm with diameter of its base 18 cm is carved out from a wooden solid sphere of radius 9 cm. The percentage of the wood wasted is :
- 45%
- 56%
- 67%
- 75%
Answer: Option D
Explanation:
We will first subtract the cone volume from wood volume to get the wood wasted.
Then we can calculate its percentage.
\begin{aligned}
\text{Sphere Volume =}\frac{4}{3}\pi r^3 \\
\text{Cone Volume =}\frac{1}{3}\pi r^2h\\
\text{Volume of wood wasted =}\\
\left(\frac{4}{3}\pi *9*9*9\right)-\left(\frac{1}{3}\pi *9*9*9\right) \\
= \pi *9*9*9 cm^3 \\
\text{Required Percentage =} \\
\frac{\pi *9*9*9}{\frac{4}{3}\pi *9*9*9}*100 \% \\
= \frac{3}{4}*100 \% \\
= 75\%
\end{aligned}
2. A cistern 6 m long and 4 m wide contains water up to a breadth of 1 m 25 cm. Find the total area of the wet surface.
- 42 m sqaure
- 49 m sqaure
- 52 m sqaure
- 64 m sqaure
Answer: Option B
Explanation:
Area of the wet surface =
2[lb+bh+hl] - lb = 2 [bh+hl] + lb
= 2[(4*1.25+6*1.25)]+6*4 = 49 m square
3. 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}
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 swimming pool 9 m wide and 12 m long is 1 m deep on the shallow side and 4 m deep on the deeper side. Its volume is:
- 260
- 262
- 270
- 272
Answer: Option C
Explanation:
Volume will be length * breadth * height, but in this case two heights are given so we will take average,
\begin{aligned}
Volume = \left(12*9*\left(\frac{1+4}{2}\right)\right)m^3 \\
12*9*2.5 m^3 = 270 m^3
\end{aligned}
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