[3 marks sum] - Mathematics STD 10 Questions - Vidyadip

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[3 marks sum]

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16 questions · timed · auto-graded

Question 13 Marks

Find the volume of the hollow sphere whose inner diameter is 8 cm and the thickness of the material of which it is made is 1 cm .

Answer

$
\begin{aligned}
& \text { Inner diameter }=8 \mathrm{~cm} \\
& \text { Inner radius }=r=4 \mathrm{~cm} \\
& \text { Outer radius }=R=4 \mathrm{~cm}+1 \mathrm{~cm} \text { thick material }=5 \mathrm{~cm} \\
& \text { Volume of hemisphere }=\frac{2}{3} \pi r^3 \\
& \text { Required Volume }=\frac{4}{3} \pi\left(R^3-r^3\right) \\
& =\frac{4}{3} \times \frac{22}{7} \times\left(5^3-4^3\right) \\
& =\frac{4}{3} \times \frac{22}{7} \times 61 \\
& =255.6 \mathrm{~cm}^3
\end{aligned}
$
Required volume $=255.6 \mathrm{~cm}^3$

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Question 23 Marks

A solid sphere metal is cut through its centre into 2 equal parts. If the diameter of the sphere is $3 \frac{1}{3} cm$, find the total surface of each part, correct to two decimal places.

Answer

Diameter of the sphere $=3 \frac{1}{3} \mathrm{~cm}=\frac{10}{3} \mathrm{~cm}$
Therefore, radius $=\frac{5}{3} \mathrm{~cm}$
Total curved surface area of each hemisphere $=3 \pi r^2$
$
\begin{aligned}
& =3 \times \frac{22}{7} \times \frac{5}{3} \times \frac{5}{3} \\
& =26.19 \mathrm{~cm}^2
\end{aligned}
$
Total curved surface area of each hemisphere $=26.19 \mathrm{~cm}^2$

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Question 33 Marks

Find the cost of painting a hemispherical dome of diameter 10 m at the rate of Rs 1.40 per square metre.

Answer

Diameter of the hemispherical dome $=10 \mathrm{~m}$
Therefore, radius of dome $=5 \mathrm{~m}$
Curved surface area $=2 \pi r^2$
$
\begin{aligned}
& =2 \times \frac{22}{7} \times 5 \times 5 \\
& =157.14 \mathrm{~m}^2
\end{aligned}
$
Cost of painting one sq. metre $=$ Rs. 1.40
Cost of painting $157.14 \mathrm{~m}^2=$ Rs. $(1.40 \times 157.14)$
$=$ Rs. $219.99=$ Rs 220
Therefore, cost of painting the dome $=$ Rs 220

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Question 43 Marks

Find the total surface area and volume of a hemisphere whose radius is 10 cm.

Answer

$
\begin{aligned}
& \text { Radius }=10 \mathrm{~cm} \\
& \text { Total surface area }=3 \pi r^2 \\
& =3 \times \frac{22}{7} \times 10 \times 10 \mathrm{~cm}^2 \\
& =942.86 \mathrm{~cm}^2
\end{aligned}
$
$
\begin{aligned}
& \text { Volume of hemisphere }=\frac{2}{3} \pi r^3 \\
& =\frac{2}{3} \times 3.14 \times 10 \times 10 \times 10 \mathrm{~cm}^3 \\
& =2093.3 \mathrm{~cm}^3
\end{aligned}
$
Total surface area $=942.86 \mathrm{~cm}^2$ and volume $=2093.3 \mathrm{~cm}^3$

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Question 53 Marks

The radius and height of a cylinder, a cone and a sphere are same. Calculate the ratio of their volumes.

Answer

Let $r , h$ be the radius and height of Cylinder, Cone and Sphere.
Volume of cylinder $=\pi r^2 h$
Volume of sphere $=\frac{4}{3} \pi r^3 \quad( h =2 r$ for sphere $)$
Volume of cone $=\frac{1}{3} \pi r^2 h$
$\pi r^2 h: \frac{1}{3} \pi r^2 h: \frac{4}{3} \pi r^3$
The volume of a cylinder is three times the volume of a cone with equal height and radius. The volume of a sphere is two times the volume of a cone with equal height and radius.
So the ratio of volumes is $3: 1: 2$.

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Question 63 Marks

Find the radius of a sphere whose surface area is equal to the area of the circle of diameter $2.8\ cm$

Answer

Diameter of circle $=2.8 cm \Rightarrow$ radius $=r=1.4 cm$
Area of a circle $=\pi r^2$
$=\pi(1.4)^2$
$=1.96 \pi$
Surface area of sphere $=4 \pi r^2$
Given ,
Surface area of sphere $=$ Area of the circle
$\Rightarrow 4 \pi r^2=1.96 \pi$
$\Rightarrow r^2=\frac{1.96}{4}$
$\Rightarrow r^2=0.49$
$\Rightarrow r=0.7 cm$
Radius of the sphere $=0.7 cm$

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Question 73 Marks

How many lead balls of radii 1 cm each can be made from a sphere of 8 cm radius?

Answer

Volume of sphere $=\frac{4}{3} \pi r^3$
Volume of sphere $=\frac{4}{3} \pi 8^3$
$=\frac{4}{3} \pi 512$
$=682.6667 \pi$
Volume of lead ball $=\frac{4}{3} \pi 1^3$
$=1.333333 \pi$
No. of lead balls that can be made $=\frac{682.6667 \pi}{1.333333 \pi}=512$
No. of lead balls that can be made $=512$

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Question 83 Marks

Find the diameter of the sphere for the following:
Surface Area $=576 \pi cm ^2$

Answer

Surface Area $=576 \pi \mathrm{cm}^2$
$
\begin{aligned}
& \text { Surface area }=4 \pi r^2 \\
& \Rightarrow 4 \pi r^2=576 \pi \\
& \Rightarrow 4 r^2=576 \\
& \Rightarrow r^2=\frac{576}{4} \\
& \Rightarrow r^2=144 \\
& \Rightarrow r=12 \mathrm{~cm}
\end{aligned}
$
Radius $=12 \mathrm{~cm} \Rightarrow$ Diameter $=2 \mathrm{r}=24 \mathrm{~cm}$

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Question 93 Marks

Find the diameter of the sphere for the following :
Surface Area = $221. 76 cm^2$

Answer

$
\begin{aligned}
& \text { Surface Area }=221.76 \mathrm{~cm}^2 \\
& \text { Surface area }=4 \pi r^2 \\
& \Rightarrow 4 \pi r^2=221.76 \\
& \Rightarrow 4 \times \frac{22}{7} \times r^2=221.76 \\
& \Rightarrow r^2=\frac{221.76 \times 7}{4 \times 22} \\
& \Rightarrow r^2=17.64 \\
& \Rightarrow r=4.2 \mathrm{~cm}
\end{aligned}
$
Radius $=4.2 \mathrm{~cm} \Rightarrow$ Diameter $=2 \mathrm{r}=8.4 \mathrm{~cm}$

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Question 103 Marks

Find the diameter of the sphere for the following:
$
\text { Volume }=72 \pi cm ^3
$

Answer

$
\begin{aligned}
& \text { Volume }=72 \pi \mathrm{cm}^3 \\
& \text { Volume of sphere }=\frac{4}{3} \pi r^3=72 \pi \\
& \Rightarrow \frac{4}{3} \pi r^3=72 \pi \\
& \Rightarrow \frac{4}{3} \times r^3=72 \\
& \Rightarrow r^3=\frac{72 \times 3}{4}=\frac{216}{4} \\
& \Rightarrow r^3=54 \\
& \Rightarrow r=3 \sqrt[3]{2} \mathrm{~cm}
\end{aligned}
$
Radius $=3 \sqrt[3]{2} \mathrm{~cm} 3 \sqrt[3]{2} \Rightarrow$ Diameter $=2 \mathrm{r}=6 \sqrt[3]{2} \mathrm{~cm} \sqrt[3]{2}$

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Question 113 Marks

Find the diameter of the sphere for the following:
$
\text { Volume }=523 \frac{17}{21} cm ^3
$

Answer

Volume $=523 \frac{17}{21} \mathrm{~cm}^3$
Volume of sphere $=\frac{4}{3} \pi r^3$
$
\begin{aligned}
& \Rightarrow \frac{4}{3} \pi r^3=523 \frac{17}{21} \\
& \Rightarrow \frac{4}{3} \times \frac{22}{7} \times r^3=\frac{11000}{21} \\
& \Rightarrow r^3=\frac{11000 \times 7 \times 3}{21 \times 4 \times 22}=\frac{231000}{1848} \\
& \Rightarrow r^3=125 \\
& \Rightarrow r=5 \mathrm{~cm}
\end{aligned}
$
Radius $=5 \mathrm{~cm} \Rightarrow$ Diameter $=2 \mathrm{r}=10 \mathrm{~cm}$

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Question 123 Marks

Find the volume and the surface area of the spheres in the following :
Diameter= 6.3 cm

Answer

Diameter $=6.3 \mathrm{~cm}$
Diameter $=6.3 \mathrm{~cm} \Rightarrow$ radius $(\mathrm{r})=3.15 \mathrm{~cm}$
Volume of sphere $=\frac{4}{3} \pi r^3$
$
\begin{aligned}
& =\frac{4}{3} \times \frac{22}{7} \times 3.15 \times 3.15 \times 3.15 \\
& =130.98 \mathrm{~cm}^3
\end{aligned}
$
Surface area $=4 \pi r^2$
$
\begin{aligned}
& =4 \times \frac{22}{7} \times 3.15 \times 3.15 \\
& =124.74 \mathrm{~cm}^2
\end{aligned}
$
Volume $=130.98 \mathrm{~cm}^3$ and Surface area $=124.74 \mathrm{~cm}^2$

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Question 133 Marks

Find the volume and the surface area of the spheres in the following :
Radius = 7 cm

Answer

Radius $=7 \mathrm{~cm}$
Volume of sphere $=\frac{4}{3} \pi r^3$
$
\begin{aligned}
& =\frac{4}{3} \times \frac{22}{7} \times 7 \times 7 \times 7 \\
& =1437.33 \mathrm{~cm}^3
\end{aligned}
$
Surface area $=4 \pi r^2$
$
=4 \times \frac{22}{7} \times 7 \times 7
$
$
=616 \mathrm{~cm}^2
$
Volume $=1437.33 \mathrm{~cm}^3$ and Surface area $=616 \mathrm{~cm}^2$

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Question 143 Marks

Find the volume and the surface area of the spheres in the following :
Radius= 2.1 cm

Answer

$
\text { Radius }=2.1 \mathrm{~cm}
$
Volume of sphere $=\frac{4}{3} \pi r^3$
$
\begin{aligned}
& =\frac{4}{3} \times \frac{22}{7} \times 2.1 \times 2.1 \times 2.1 \\
& =38.808 \mathrm{~cm}^3
\end{aligned}
$
Surface area $=4 \pi r^2$
$
\begin{aligned}
& =4 \times \frac{22}{7} \times 2.1 \times 2.1 \\
& =55.44 \mathrm{~cm}^2
\end{aligned}
$
Volume $=38.808 \mathrm{~cm}^3$ and Surface area $=55.44 \mathrm{~cm}^2$

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Question 153 Marks

The curved surface area of a right circular cone of radius $11.3 cm$ is $710 cm^2$. What is the slant height of the cone ?

Answer

Curved surface area $=710 \mathrm{~cm}^2$
Radius (r) of base $=11.3 \mathrm{~cm}$
Let Slant height be I.
$
\begin{aligned}
& \therefore \pi r l=710 \\
& \Rightarrow \frac{22}{7} \times 11.3 \times l=710 \\
& \Rightarrow l=\frac{710 \times 7}{11.3 \times 22} \\
& \Rightarrow I=19.99 \mathrm{~cm}=20 \mathrm{~cm}
\end{aligned}
$
The slant height is $20 \mathrm{~cm}$.

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Question 163 Marks

The base circumferences of two cones are the same. If their slant heights are in the ratio 5 : 4, find the ratio of their curved surface areas.

Answer

The base circumferences of the cones are equal, therefore the radius of base are equal.
Let radius be $\mathrm{r}$.
Ratio between slant heights $=5: 4$
Let slant height of first cone $=5 \mathrm{x}$ and of second cone $=4 \mathrm{x}$
Curved surface area of cone $=\pi r l$ (where $\mathrm{I}=$ slant height)
Ratio of curved surface areas $=\frac{\pi r \times 5 x}{\pi r \times 4 x}$
$=\frac{5}{4}$
Ratio of curved surface areas $=5: 4$

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[3 marks sum] - Mathematics STD 10 Questions - Vidyadip