Question 12 Marks
The inner diameter of a well is 4.20 metre and its depth is 10 metre. Find the inner surface area of the well. Find the cost of plastering it from inside at the rate ₹ 52 per sq.m.
Given: Inner diameter (d) = 4.2 m,
To find: depth (h) = 10 m,
rate of plastering = ₹ 52 per sq.m.
Inner surface area and total cost of plastering
Answer$\text { i. Inner curved surface area of the well }=2 \pi \text { rh }$
$=\pi d h \ldots[\because d=2 r]$
$=\sqrt[22]{7} \times 4.2 \times 10$
$=\sqrt[22]{7} \times 42$
$=22 \times 6$
$=132 \text { sq.m. }$
ii. Rate of plastering $= ₹52$ per sq.m.
$\therefore$ Total cost = Curved surface area x Rate of plastering
$= 132 \times 52 = ₹6864$
$\therefore$ The cost of plastering the well from inside is $₹6864.$
View full question & answer→Question 22 Marks
Total surface area of a cone is 616 sq.cm. If the slant ‘height of the cone Is three times the radius of its base, find its slant height.
Given: Total surface area of a cone = 616 sq.cm., slant height of the cone is three times the radius of its base
To find: Slant height (l)
Answeri. Let the radius of base be r cm.
$\therefore$ Slant height $(l) = 3r cm$
Total surface area of cone $= \pi r (l + r)$
$\therefore 616 = \pi r(l + r)$
$\therefore 616=\sqrt[22]{7} \times r \times(3 r+r)$
$\therefore 616=\sqrt[22]{7} \times 4 r^2$
$\therefore \quad r^2=\frac{616 \times 7}{22 \times 4}$
$=\frac{28 \times 7}{4}$
$\therefore r^2=49$
$\therefore r=\sqrt{49} \text {... [Taking square root on both sides] }$
$=7$
ii. Slant height $(l) = 3r = 3 \times 7 = 21 cm$
$\therefore$ The slant height of the cone is $21 cm.$
View full question & answer→Question 32 Marks
Total surface area of a cube is 864 sq.cm. Find its volume.
Given: Total surface area of cube $=864 sq . cm$
To find: Volume of cube
Answeri. Total surface area of cube $=61^2$
$\therefore 864=61^2$
$\therefore 1^2=\sqrt[864]{6}$
$\therefore 1^2=144$
$\therefore 1=\sqrt{144} \ldots$ [Taking square root on both sides]
$=12 cm$
ii. Volume of cube $=1^2$
$=12^3$
$=1728$ cubic cm.
$\therefore$ The volume of cube is 1728 cubic cm .
View full question & answer→Question 42 Marks
Find the radius of a sphere if its volume is 904.32 cubic cm. (π = 3.14)
Given: Volume of sphere = 904.32 cubic cm.
To find: Radius of a sphere
Answer$\text { Volume of sphere }=\frac{4}{3} \pi r^3$
$\therefore 904.32=\frac{4}{3} \times 3.14 \times r^3$
$ \therefore \quad r^3 =\frac{904.32 \times 3}{4 \times 3.14}$
$ =\frac{90432 \times 3}{4 \times 314}$
$ =\frac{288 \times 3}{4}$
$= 216$
$\therefore r =\sqrt[3]{216} \ldots$ [Taking cube root on both sides]
$= 6\ cm$
$\therefore $ The radius of the sphere is $6\ cm.$
View full question & answer→Question 52 Marks
Find the volume of a sphere whose surface area is 314.sq.cm. (Take p = 3.14)
AnswerSurface area of sphere $=4 \pi r^2$
$
\begin{aligned}
314 & =4 \times 3.14 \times r^2 \\
\frac{314}{4 \times 3.14} & =r^2 \\
\frac{31400}{4 \times 314} & =r^2 \\
\therefore \quad \frac{100}{4} & =r^2 \\
\therefore \quad 25 & =r^2 \\
\therefore \quad r & =5 \mathrm{~cm}
\end{aligned}
$$\begin{aligned} \text { Volume of sphere } & =\frac{4}{3} \pi r^3 \\ & =\frac{4}{3} \times 3.14 \times 5^3 \\ & =\frac{4}{3} \times 3.14 \times 125 \\ & =523.33 \text { cubic cm. }\end{aligned}$
View full question & answer→Question 62 Marks
Area of the base of a cone is $1386$ sq. $\mathrm{cm}$ and its height is $28 \mathrm{~cm}$.Find its surface area. $\left(\pi=\frac{22}{7}\right)$
AnswerArea of base of cone $=\pi r^2$
$ \therefore 1386 =\frac{22}{7} \times r^2$
$ \therefore \frac{1386 \times 7}{22} =r^2$
$ \therefore 63 \times 7 =r^2$
$ \therefore 441 =r^2$
$ \therefore r =21 \mathrm{~cm} $
$ \therefore l^2=(21)^2+(28)^2$
$\therefore l^2=441+784$
$\therefore l^2=1225$
$\therefore l=35 \mathrm{~cm} $
Surface area of cone $=\pi \mathrm{rl}$
$ =\frac{22}{7} \times 21 \times 35$
$=22 \times 21 \times 5$
$=2310 \text { sq. cm. } $
View full question & answer→Question 72 Marks
If the radius of a solid hemisphere is 5 cm , then find its curved surface area and total surface area, ( $\pi=3.14$ )
Given: Radius $(r)=5 cm$
To find: Curved surface area and total surface area of hemisphere
Answeri. Curved surface area of hemisphere $=2 \pi r ^2$
$=2 \times 3.14 \times 5^2$
$=2 \times 3.14 \times 25$
$=50 \times 3.14$
$=157 sq.cm.ii. Total surface area of hemisphere=3 \pi r^2$
$=3 \times 3.14 \times 5^2$
$=235.5 sq . cm .$
$\therefore$ The curved surface area and totai surface area of hemisphere are $157 sq . cm$, and $235.5 sq . cm$, respectively.
View full question & answer→Question 82 Marks
Find the surface areas and volumes of spheres of the following radii : $3.5 cm(\pi=3.14)$
AnswerGiven: Radius $( r )=3.5 cm$
To find: Surface area and volume of sphere
Solution:
Surface area of sphere $=4 \pi r ^2$
$=4 \times 3.14 \times(3.5)^2$
$\therefore$ Surface area of sphere $=153.86 sq \cdot cm$
Volume of sphere $=\frac{4}{3} \pi r^3$ $=\frac{4}{3} \times 3.14 \times(3.5)^3$
$\therefore$ Volume of sphere $=179.50$ cubic $cm$
View full question & answer→Question 92 Marks
Find the surface areas and volumes of spheres of the following radii : $9 cm$
AnswerGiven: Radius $(r) = 9 cm$
To find: Surface area and volume of sphere
Solution:
Surface area of sphere = $4\pi r^2$
$= 4 x 3.14 x 9^2$
$\therefore Surface area of sphere = 1017.36 sq.cm$
$
\begin{aligned}
\text { Volume of sphere } & =\frac{4}{3} \pi r ^3 \\
& =\frac{4}{3} \times 3.14 \times 9^3 \\
& =\frac{4}{3} \times 3.14 \times 9 \times 9 \times 9 \\
& =4 \times 3.14 \times 3 \times 9 \times 9
\end{aligned}
$
$\therefore$ Volume of sphere $=3052.08$ cubic $cm$
View full question & answer→Question 102 Marks
Find the surface areas and volumes of spheres of the following radii : $4 cm$
AnswerGiven: Radius ( $r$ ) $=4 cm$
To find: Surface area and volume of sphere
Surface area of sphere $=4 \pi r^2$
$=4 \times 3.14 \times 4^2$
$\therefore$ Surface area of sphere $=200.96 sq . cm$
Volume of sphere $=\frac{4}{3} \pi r^3=\frac{4}{3} \times 3.14 \times 4^2$
$\therefore$ Volume of sphere $=267.95$ cubic cm
View full question & answer→Question 112 Marks
The curved surface area of a cone is $2200$ sq.cm and its slant height is $50\ cm.$ Find the total surface area of cone.$\left(\pi=\frac{22}{7}\right)$
Given: Length $(l) = 50\ cm,$ curved surface area of cone $= 2200$ sq.cm
To find: Total surface area of the cone
Answeri. Curved surface area of cone $= \pi rl$
$\therefore 2200=\frac{22}{7} \times r \times 50 $
$ \therefore r =\frac{2200 \times 7}{22 \times 50}$
$=\frac{100 \times 7}{50}=14 cm$
ii. Total surface area of cone $= \pi r (l + r)$
$ =\frac{22}{7} \times 14 \times(50+14)$
$=\frac{22}{7} \times 14 \times 64$
$=22 \times 2 \times 64$
$=2816\ sq .\ cm $
$\therefore$ The total surface area of the cone is $2816\ sq .\ cm$.
View full question & answer→Question 122 Marks
Volume of a cone is $6280$ cubic cm and base radius of the cone is $20\ cm.$ Find its perpendicular height, $(\pi = 3.14)$
Given: Radius $(r) = 20\ cm,$
Volume of cone $= 6280$ cubic cm
To find: Perpendicular height $(h)$ of the cone
Answer$ \text { Volume of cone }=\frac{1}{3} \pi r^2 h$
$\therefore \quad 6280=\frac{1}{3} \times 3.14 \times 20^2 \times h$
$\therefore \quad h =\frac{6280 \times 3}{3.14 \times 400}$
$=\frac{6280 \times 3}{314 \times 4}$
$=\frac{20 \times 3}{4}=15 cm$
$\therefore$ The perpendicular height of the cone is $15 cm$.
View full question & answer→Question 132 Marks
What will be the cost of making a closed cone of tin sheet having radius of base $6\ m$ and slant height $8\ m$ if the rate of making is ₹ $10$ per sq.m?
Given: Radius $(r) = 6 m,$ length $(l) = 8 m$
To find: Total cost of making the cone
Answeri. To find the total cost of making the cone of tin sheet, first we need to find the total surface area of the cone.
Total surface area of the cone $= \pi r (l + r)$
$=\frac{22}{7} \times 6 \times(8+6)$
$=\frac{22}{7} \times 6 \times 14$
$=22 \times 6 \times 2=264 \text { sq.m }$
ii. Rate of making the cone $= ₹ 10$ per sq.m
$\therefore$ Total cost $=$ Total surface area x Rate of making the cone
$= 264 \times 10$
$= ₹ 2640$
$\therefore$ A The total cost of making the cone of tin sheet is ₹ $2640.$
View full question & answer→Question 142 Marks
Perpendicular height of a cone is 12 cm and its slant height is 13 cm . Find the radius of the base of the cone.
Given: Height $( h )=12 cm$, length $( l )=13 cm$
To find: Radius of the base of the cone ( r )
Answer$r^2=r^2+h^2$
$\therefore 13^2=r^2+12^2$
$\therefore 169=r^2+144$
$\therefore 169-144=r^2$
$\therefore r^2=25$
$\therefore r=\sqrt{ } 25 \ldots \text { [Taking square root on both sides] }$
$=5 cm$
$\therefore$ The radius of base of the cone is 5 cm .
View full question & answer→Question 152 Marks
Curved surface area of a cylinder is $1980 cm ^2$ and radius of its base is $15 cm$. Find the height of the cylinder. $\left(\pi=\frac{22}{7}\right)$
Given: Curved surface area of cylinder $=1980$ sq.cm., radius $(r)=15 cm$
To find: Height of the cylinder (h)
AnswerCurved surface area of cylinder $=2 \pi r h$
$ \therefore 1980=2 \times \frac{22}{7} \times 15 \times h$
$\therefore h=\frac{1980 \times 7}{2 \times 22 \times 15}$
$\therefore h =21 cm $
$\therefore$ The height of the cylinder is $21 cm$.
View full question & answer→Question 162 Marks
Radius of base of a cylinder is 20 cm and its height is 13 cm, find its curved surface area and total surface area, (π = 3.14)
Given: Radius (r) = 20 cm, height (h) = 13 cm
To find: Curved surface area and
the total surface area of the cylinder
Answeri. Curved surface area of cylinder = 2πrh
= 2 x 3.14 x 20 x 13
= 1632.8 sq.cm
ii. Total surface area of cylinder = 2πr(r + h)
= 2 x 3.14 x 20(20 + 13)
= 2 x 3.14 x 20 x 33 = 4144.8 sq.cm
∴ The curved surface area and the total surface area of the cylinder are 1632.8 sq.cm and 4144.8 sq.cm respectively.
View full question & answer→Question 172 Marks
What will be the volume of a cube having length of edge 7.5 cm ?
Given: Length of edge of cube $(1)=7.5 cm$
To find: Volume of a cube
AnswerVolume of a cube $=1^2$
$=(7.5)^3$
$=421.875 \approx 421.88$ cubic cm
$\therefore$ The volume of the cube is 421.88 cubic cm .
View full question & answer→Question 182 Marks
Volume of a cuboid is $34.50$ cubic metre. Breadth and height of the cuboid is $1.5\ m$ and $1.15\ m$ respectively. Find its length.
Given: Breadth $(b) = 1.5\ m,$ height $(h) = 1.15\ m$
Volume of cuboid $= 34.50$ cubic metre
To find: Length of the cuboid (l)
AnswerVolume of cuboid $=1 \times b \times h$
$ \therefore 34.50=1 \times b \times h$
$\therefore 34.50=1 \times 1.5 \times 1.15$
$\therefore \quad l=\frac{34.50}{1.5 \times 1.15}$
$=\frac{34500}{15 \times 115}$
$=\frac{300}{15}$
$=20$
$=20 $
$\therefore$ The length of the cuboid is $20 m$.
View full question & answer→Question 192 Marks
Total surface area of a cube is $5400$ sq. cm. Find the surface area of all vertical faces of the cube.
Given: Total surface area of cube $= 5400$ sq.cm.
To find: Surface area of all vertical faces of the cube
Answeri. Total surface area of cube $=\left.6\right|^2$
$ \therefore 5400=6 I ^2$
$\therefore \frac{5400}{6}= R ^2$
$\therefore R^2=900 $
ii. Area of vertical faces of cube $=41^2$
$=4 \times 900=3600 sq . cm \text {. }$
$\therefore$ The surface area of all vertical faces of the cube is $3600\ sq.cm.$
View full question & answer→Question 202 Marks
Side of a cube is 4.5 cm . Find the surface area of all vertical faces and total surface area of the cube.
Given: Side of cube $(I)=4.5 cm$
To find: Surface area of all vertical faces and the total surface area of the cube
Answeri. Area of vertical faces of cube $=41^2$
$=4(4.5)^2=4 \times 20.25=81 sq . cm .$
ii. Total surface area of the cube $=61^2$
$=6(4.5)^2$
$=6 \times 20.25$
$=121.5 sq . cm .$
$\therefore$ The surface area of all vertical faces and the total surface area of the cube are $81 sq.cm$, and $121.5 sq . cm$, respectively.
View full question & answer→Question 212 Marks
Total surface area of a box of cuboid shape is $500$ sq.unit. Its breadth and height is $6$ unit and $5$ unit respectively. What is the length of that box?
Given: For cuboid shape box,
breadth $(b) = 6$ unit, height $(h) = 5$ unit Total surface area $= 500$ sq. unit.
To find: Length of the box $(l)$
AnswerTotal surface area of the box $=2( lb + bh + lh )$
$ \therefore 500=2(61+6 \times 5+51)$
$\therefore \frac{500}{2}=(11 \mid+30)$
$\therefore 250=11 \mid+30$
$\therefore 250-30=11 \mid$
$\therefore 220=11 \mid$
$\therefore 220=1$
$\therefore \frac{220}{11}=1$
$\therefore I=20 \text { units } $
$\therefore$ The length of the box is $20$ units.
View full question & answer→Question 222 Marks
Length, breadth and height of a cuboid shape box of medicine is 20 cm, 12 cm and 10 cm respectively. Find the surface area of vertical faces and total surface area of this box.
Given: For cuboid shape box of medicine,
length (l) = 20 cm, breadth (b) = 12 cm and height (h) = 10 cm.
To find: Surface area of vertical faces and total surface area of the box
Answeri. Surface area of vertical faces of the box
= 2(l + b) x h
= 2(20+ 12) x 10
= 2 x 32 x 10
= 640 sq.cm.
ii. Total surface area of the box
= 2 (lb + bh + lh)
= 2(20 x 12+ 12 x 10 + 20 x 10)
= 2(240 + 120 + 200)
= 2 x 560
= 1120 sq.cm.
∴ The surface area of vertical faces and total surface area of the box are 640 sq.cm, and 1120 sq.cm, respectively.
View full question & answer→