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Surface Area And Volume of A Right Circular Cylinder · MP Board (MPBSE) STD 9 (Madhya Pradesh - English Medium). 20 questions.

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MCQ 11 Mark
In a cylinder, if radius is doubled and height is halved, curved surface area will be:
  • A
    Halved.
  • B
    Doubled.
  • ✓
    Same.
  • D
    Four times.
Answer
Correct option: C.
Same.
curved surface area of a cylinder of radius $'r' $and heightn $'h' $is given by $\text{A}=2\pi\text{rh}$
Now, if $r' = 2r $and $\text{h}'=\frac{\text{h}}{2}$
Then $\text{A}'=2\pi\times(2\text{r})\times\frac{\text{h}}{2}$
$=2\pi\text{rh}=\text{A}$
$\Rightarrow C.S.A.$ remains the same.
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MCQ 21 Mark
The height h of a cylinder equals the circumference of the cylinder. In terms of h what is the volume of the cylinder?
  • ✓
    $\frac{\text{h}^3}{4\pi}$
  • B
    $\frac{\text{h}^2}{2\pi}$
  • C
    $\frac {\text{h}^3}{2}$
  • D
    $\pi\text{h}^3$
Answer
Correct option: A.
$\frac{\text{h}^3}{4\pi}$
Circumference of cylinder $=2\pi\text{r}$
Height $= h$
Volume $\pi\text{r}^2\text{h}=\pi\Big(\frac{\text{h}^2}{4\pi^2}\Big)\text{h}=\frac{\text{h}^3}{4\pi}$
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MCQ 31 Mark
If the radius of a cylinder is doubled and the height remains same, the volume will be:
  • A
    Doubled.
  • B
    Halved.
  • C
    Same.
  • ✓
    Four times.
Answer
Correct option: D.
Four times.
Volume of a cylinder $=\text{V}=\pi\text{r}^2\text{h}$
If $\text{r}'=2\text{r} $ and $\text{h}'=\text{h} $ then
$\text{V}'=\pi(2\text{r})^2\text{h}=4\pi\text{r}^2\text{h}$
$\text{V}'=4\text{V}$
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MCQ 41 Mark
Two circular cylinders of equal volume have their heights in the ratio $1 : 2$ Ratio of their radii is:
  • A
    $1:\sqrt{2}$
  • ✓
    $\sqrt{2}:1$
  • C
    $1:2$
  • D
    $1:4$
Answer
Correct option: B.
$\sqrt{2}:1$

Volume of cylinder 1, $\text{v}_1=\pi\text{r}^2_1\text{h}_1$
Volume of cylinder 1, $\text{v}_2=\pi\text{r}^2_2\text{h}_2$
$\frac{\text{v}_1}{\text{v}_2}=\frac{\text{r}^2_1}{\text{r}^2_2}\frac{\text{h}_1}{\text{h}_2}...(1)$
Now, $v_1 = v_2$ and $\frac{\text{h}_1}{\text{h}_2}=\frac{1}{2}$
Hence, Equation (1) reduces to
$1=\frac{\text{r}^2_1}{\text{r}^2_2}=\frac{1}{2}$
$\Rightarrow\frac{\text{r}^2_2}{\text{r}^2_1}=\frac{1}{2}$
$\Rightarrow\frac{\text{r}^2_1}{\text{r}^2_2}=2$
$\Rightarrow\frac{\text{r}^2_1}{\text{r}^2_2}=\frac{\sqrt{2}}{1}$
$\Rightarrow\text{r}_1:\text{r}_2=\sqrt{2}:1$

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MCQ 51 Mark
The height of sand in a cylindrical shaped can drops $3$ inches when $1$ cubic foot of sand is poured out. What is the diameter, in inches, of the cylinder?
  • A
    $\frac{24}{\sqrt{\pi}}$
  • ✓
    $\frac{48}{\sqrt{\pi}}$
  • C
    $\frac{32}{\sqrt{\pi}}$
  • D
    $\frac{48}\pi$
Answer
Correct option: B.
$\frac{48}{\sqrt{\pi}}$
When sand is poured out, height dropped = 3 inches
Volume vacant $=\pi\text{r}^2\times3\text{ inches}$
Now, Volume vacant = Volume of sand poured out $= 1$ cubic foot
$1$ foot $= 12$ inches
$1$ cubic foot $= 12 \times 12 \times 12$ inches $= 1728$ inches
Thus, We have
$3\pi\text{r}^2=1728$
$\Rightarrow\pi\text{r}^2=576$
$\Rightarrow\text{r}^2=\frac{576}{\pi}$
$\Rightarrow\text{r}=\frac{24}{\sqrt{\pi}}$
$\Rightarrow $ Diameter $= 2r $$=2\times\frac{24}{\sqrt{\pi}}=\frac{48}{\sqrt{\pi}}$
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MCQ 61 Mark
The number of surfaces in right cylinder is:
  • A
    $1$
  • B
    $2$
  • ✓
    $3$
  • D
    $4$
Answer
Correct option: C.
$3$
Number of Surfaces In a Right cylinder are $3$.
Top surface, bottom surface and curved surface.
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MCQ 71 Mark
Vertical cross-section of a right circular cylinder is always a:
  • A
    Square.
  • ✓
    Rectangle.
  • C
    Rhombus.
  • D
    Trapezium.
Answer
Correct option: B.
Rectangle.
Vertical cross-section of cylinder will always be a Rectangle of sides $'h'$, and$ 'r',$
where $h$ is the height of a cylinder and $r$ is the radius of a cylinder
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MCQ 81 Mark
A right circular cylindrical tunnel of diameter $2m$ and length $40m$ is to be constructed from a sheet of iron. The area of the iron sheet required in $m^2$, is:
  • A
    $40\pi$
  • ✓
    $80\pi$
  • C
    $160\pi$
  • D
    $200\pi$
Answer
Correct option: B.
$80\pi$

Cylinderical tunnel will be hollow cylinder of radius $= 1m$
$\big\{\text{r}=\frac{\text{d}}{2}=\frac{2}{2}=1\text{m}\big\}$
Length $= 40\ m$
Area of iron sheet = Curved surface area of cylinder
$=2\pi\text{rh}$
$=2\pi(1)40$
$=80\pi$

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MCQ 91 Mark
A cylinder with radius r and height h is closed on the top and bottom. Which of the following expressions represents the total surface area of this cylinder?
  • ✓
    $2\pi\text{r}(\text{r}+\text{h})$
  • B
    $\pi\text{r}(\text{r}+\text{2h})$
  • C
    $\pi\text{r}(\text{2r}+\text{h})$
  • D
    $2\pi\text{r}^2+\text{h}$
Answer
Correct option: A.
$2\pi\text{r}(\text{r}+\text{h})$
Total surface Area = Area of top + Area of bottom + Curved surface area
$T.S.A.$ $=\pi\text{r}^2+\pi\text{r}^2+2\pi\text{rh}=2\pi\text{ r}^2=2\pi\text{r}(\text{r}+\text{h})$
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MCQ 101 Mark
The number of surfaces of a hollow cylindrical object is:
  • A
    $1$
  • B
    $2$
  • C
    $3$
  • ✓
    $4$
Answer
Correct option: D.
$4$
In a hollow cylinder, there are two curved surface areas: inner and outer and one circular base with inner and outer surface area.
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MCQ 111 Mark
The radius of a wire is decreased to one-third. If volume remains the same, the length will become:
  • A
    $3$ times
  • B
    $6$ times
  • ✓
    $9$ times
  • D
    $27$ times
Answer
Correct option: C.
$9$ times
Volume of wire $=\pi\text{r}^2$
If volume remains same, then
$\pi\text{r}^2_1\text{l}_1=\pi\text{r}^2_2\text{l}_2$
If $\text{r}_2=\frac{\text{r}_1}{3},$ then
$\not\pi\text{r}^2_1\text{l}_1=\not\pi\Big(\frac{\text{r}_1}{3}\Big)^2\text{l}_2$
$\Rightarrow\text{l}_2=9\text{l}_1$
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MCQ 121 Mark
Two steel sheets each of length $a_1$ and breadth $a_2$ are used to prepare the surfaces of two right circular cylinders - one having volume $\mathrm{v}_1$ and height $\mathrm{a}_2$ and other having volume $\mathrm{v}_2$ and height $\mathrm{a}_1$. Then:
  • A
    $\text{v}_1=\text{v}_2$
  • B
    $\text{a}_1\text{v}_1=\text{a}_2\text{v}_2$
  • ✓
    $\text{a}_1\text{v}_1=\text{a}_1\text{v}_2$
  • D
    $\frac{\text{v}_1}{\text{a}_1}=\frac{\text{v}_2}{\text{a}_2}$
Answer
Correct option: C.
$\text{a}_1\text{v}_1=\text{a}_1\text{v}_2$
Surface area of both cylinder is going to be same because same sheet is used.
S.$A=a_1 a_2$
Surface area of cylinder is same
$\text{S}_1=(2\pi\text{r}_1)\text{a}_2=\text{a}_1\text{a}_2...(1)$
$\text{S}_2=(2\pi\text{r}_2)\text{a}_2=\text{a}_1\text{a}_2...(2)$
From equation $(1)$ & $(2)$
$2\pi\text{r}_1=\text{a}_1$ and $2\pi\text{r}_2=\text{a}_2$
Volume of cylinder 1, $\text{v}_1=(\pi\text{r}_1^2)\text{a}_2...(3)$
Volume of cylinder 2, $\text{v}_2=(\pi\text{r}_2^2)\text{a}_1...(4)$
Divinding equation $(3)$ by equation (4)
$\frac{\text{V}_1}{\text{V}_2}=\frac{\text{r}_1^2}{\text{r}_2^2}\frac{\text{a}_2}{\text{a}_1}=\frac{\Big(\frac{\text {a}_1}{\not2\pi}\Big)^2}{\Big(\frac{\text{a}_2}{\not2\pi}\Big)^2}\frac{\text{a}_2}{\text{a}_1}$
$=\frac{\text{a}_1\not2}{\text{a}_2\not2}\times \frac{\not\text{a}_2^2}{\not\text{a}_1}=\frac{\text{a}_1}{\text{a}_2}$
$\Rightarrow\text{a}_2\text{v}_1=\text{a}_1\text{v}_2$
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MCQ 131 Mark
If the diameter of the base of a closed right circular cylinder be equal to its height $h$, then its whole surface area is:
  • A
    $2\pi\text{h}^2$
  • ✓
    $\frac{3}{2}\pi\text{h}^2$
  • C
    $\frac{4}{3}\pi\text{h}^2$
  • D
    $\pi\text{h}^2$
Answer
Correct option: B.
$\frac{3}{2}\pi\text{h}^2$
Diameter $= 2r = h$ (Given)
$\Rightarrow\text{r}=\frac{\text{h}}{2}$
its surface area $=2\pi\text{r}(\text{h}+\text{r})$
$\text{s}=2\pi\text{r}(\text{h}+\text{r})=\not2\pi\frac{\text{h}}{\not2}\Big(\text{h}+\frac{\text{h}}{2}\Big)$
$=\pi\text{h}\Big(\frac{3}{2}\text{h}\Big)=\frac{3}{2}\pi\text{h}^2$
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MCQ 141 Mark
If r is the radius and h is height of the cylinder the volume will be:
  • A
    $\frac{1}{3}\pi\text{r}^2\text{h}$
  • ✓
    $\pi\text{r}^2\text{h}$
  • C
    $2\pi\text{r}(\text{h}+\text{r})$
  • D
    $2\pi\text{r}\text{h}$
Answer
Correct option: B.
$\pi\text{r}^2\text{h}$
Volume of cylinder
= Area of Base $\times $ Height
$=(\pi\text{r}^2)\times\text{h}$
$\text{V}=\pi\text{r}^2\text{h}$
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MCQ 151 Mark
The volume of a cylinder of radius r is $\frac{1}{4}$ of the volume of a rectangular box with a square base of side length $x$. If the cylinder and the box have equal heights, what is r in terms of $x$?
  • A
    $\frac{\text{x}^2}{2\pi}$
  • ✓
    $\frac{\text{x}}{2\sqrt{\pi}}$
  • C
    $\frac {\sqrt{2}\text{x}}{\pi}$
  • D
    $\frac{\pi}{2\sqrt{\text{x}}}$
Answer
Correct option: B.
$\frac{\text{x}}{2\sqrt{\pi}}$
Area of base of cylinder $=\pi\text{r}^2$
Area of base of box $=\text{x}^2$
Let the height of both objects = h
Then, $\text{v}_\text{cylinder}=\pi\text{r}^2\text{h}$
$\text{v}_\text{box}=\text{x}^2\text{h}$
Now, $\text{v}_\text{cylinder}=\frac{1}{4}\text{v}_\text{box}$
$\Rightarrow\pi\text{r}^2\not\text{h}=\frac{1}{4}\text{x}^2\not\text{h}$
$\Rightarrow\text{r}^2=\frac{\text{x}^2}{4\pi}$
$\Rightarrow\text{r}=\frac{\text{x}}{2\sqrt{\pi}}$
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MCQ 161 Mark
In a cylinder, if radius is halved and height is doubled, the volume will be:
  • A
    Same.
  • B
    Doubled.
  • ✓
    Halved.
  • D
    Four times.
Answer
Correct option: C.
Halved.
Volume of cylinder $\text{V}=\pi\text{r}^2\text{h}$
If $\text{r}'=\frac{\text{r}}{2}$ and $\text{h}'=2\text{h}$ then
Then $\text{V}'=\pi\Big(\frac{\text{r}}{2}\Big)^22\text{h}$
$=\frac{\pi\text{r}^2}{\not4_2}\times\not2\text{h}$
$\frac{\pi\text{r}^2\text{h}}{2}=\frac{\text{V}}{2}$
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MCQ 171 Mark
The altitude of a circular cylinder is increased six times and the base area is decreased one-ninth of its value. The factor by which the lateral surface of the cylinder increases, is:
  • A
    $\frac{2}{3}$
  • B
    $\frac{1}{2}$
  • C
    $\frac{3}{2}$
  • ✓
    $2$
Answer
Correct option: D.
$2$
If h is initial altitude, then $h' = 6h$
initial Base Area $=\pi\text{r}^2=\text{B}$
New base Area $=\text{B}'=\pi\text{r}'^2$
Now, $\text{B}'=\frac{\text{B}}{9}$
$\Rightarrow\pi\text{r}'^2=\frac{\pi\text{r}^2}{9}$
$\Rightarrow\text{r}'^2=\frac{\text{r}^2}{9}$
$\Rightarrow\text{r}'=\frac{\text{r}}{3}$
Intial Lateral surface Area $=2\pi\text{rh}$
New Lateral surface Area $=2\pi\text{r}'\text{h}'$
$=2\pi\Big(\frac{\text{r}}{3}\Big)6\text{h}$
$=2(2\pi\text{rh})$
$= 2$(Intial Lateral surface Area)
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MCQ 181 Mark
Two cylindrical jars have their diameters in the ratio $3 : 1$, but height $1 : 3$. Then the ratio of their volumes is:
  • A
    $1 : 4$
  • B
    $1 : 3$
  • ✓
    $3 : 1$
  • D
    $2 : 5$
Answer
Correct option: C.
$3 : 1$

Volume of any cylinder $=\pi\text{r}^2\text{h}$
$\text{r}=\frac{\text{d}}{2}$
if $d_1: d_2=3: 1$ then, $r_1: r_2=3: 1$
$h_1: h_2=1: 3$
Now,
$\frac{\text{V}_1}{\text{V}_2}=\frac{\pi(\text{r}_1)^2\text{h}_1}{\pi(\text{r}_2)^2\text{h}_2}=\Big(\frac{\text{r}_1}{\text{r}_2}\Big)^2$
$\frac{\text{h}_1}{\text{h}_2}=\Big(\frac{3}{1}\Big)^2\Big(\frac{1}{3}\Big)=\frac{3}{1}$
$=3:1$

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MCQ 191 Mark
If the height of a cylinder is doubled and radius remains the same, then volume will be:
  • ✓
    Doubled.
  • B
    Halved.
  • C
    Same.
  • D
    Four times.
Answer
Correct option: A.
Doubled.
Volume of cylinder $\text{V}=\pi\text{r}^2\text{h}$
If $\text{h}'=2\text{h} $ and $\text{r}'=\text{r},$ then
$\text{V}'=\pi\text{(r)}^2(2\text{h})$
$=2\pi\text{r}^2\text{h}=2\text{V}$
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MCQ 201 Mark
If the height of a cylinder is doubled, by what number must the radius of the base be multiplied so that the resulting cylinder has the same volume as the original cylinder?
  • A
    $4$
  • ✓
    $\frac{1}{\sqrt{2}}$
  • C
    $2$
  • D
    $\frac{1}{2}$
Answer
Correct option: B.
$\frac{1}{\sqrt{2}}$
Volume of a cylinder $=\text{v}=\pi\text{r}^2\text{h}$
Now, if $h' = 2h$ and new radius $= r'$, then
$\text{v}'=\pi\text{r}'^2\text{h}'=\pi\text{r}'^2(2\text{h})$
$=2\pi\text{r}'^2\text{h}$
Now if volume should remain same, then
$\text{v}'=\text{v}$
$\Rightarrow2\not\pi\text{r}'^2\not\text{h}=\not\pi\text{r}^2\not\text{h}$
$\Rightarrow2\text{r}'^2=\text{r}^2$
$\Rightarrow\text{r}'^2=\frac{\text{r}^2}{2}$
$\Rightarrow\text{r}'^=\frac{\text{r}}{\sqrt{2}}$
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