Question 14 Marks
Find the area of canvas required for a conical tent of height 24m and base radius $7m.$
AnswerThe amount of canvas required to make a cone would be equal to the curved surface area of the cone.
The formula of the curved surface area of a cone with base radius $‘r’$ and slant height $‘l’$ is given as:
Curved Surface Area $=\pi\text{rl}$ It is given that the vertical height $‘h’ = 24\ m$ and base radius $‘r’ = 7\ m.$
To find the slant height ‘l’ we use the following relation Slant height, $\text{l}=\sqrt{\text{r}^2+\text{h}^2}$
$\text{l}=\sqrt{7^2+24^2}$
$=\sqrt{49+576}$
$=\sqrt{625}$
$\text{l}=25$
Hence the slant height of the given cone is $25m$.
Now, substituting the values of $r = 7\ m$ and slant height $l = 25m$ and using $\pi=\frac{22}{7}$ in the formula of
$C.S.A$, We get Curved Surface Area $=\frac{(22).(7).(25)}{7}$ $= (22)(25) = 550$
Therefore the Curved Surface Area of the cone is $550m^2.$
View full question & answer→Question 24 Marks
Monica has a piece of Canvas whose area is$ 551\ m^2$. She uses it to have a conical tent made, with a base radius of $7\ m.$ Assuming that all the stitching margins and wastage incurred while cutting amounts to approximately $1\ m^2$. Find the volume of the tent that can be made with it.
AnswerIt is given that:
Area of the canvas $=551 m^2$
Area that is wasted $=1 m^2$
Radius of tent $=7 m$, Volume of tent $( v )=$ ?
Therefore the Area of available for making the tent $= (551 - 1) = 550m^2$
Surface area of tent $= 550m^2$
$\Rightarrow\pi\text{rl}=550$
$\Rightarrow\text{l}=\frac{550}{22}=25\text{m}$
Slant height $(l) = 25\ m$
We know that,$ l^2 = r^2 + h^2 25^2 = 7^2 + h^2$
$\Rightarrow 625 - 49 = h^2$
$\Rightarrow 576 = h^2 h = 24m$
Height of the tent is 24m.
Now, volume of cone $=\frac{1}{3}\pi\text{r}^2\text{h}$
$=\frac{1}{3}\times3.14\times7^2\times24=1232\text{m}^3$
Therefore the volume of the conical tent is $1232m^3.$
View full question & answer→Question 34 Marks
Find the area of metal sheet required in making a closed hollow cone of base radius $7\ cm$ and height $24\ cm$.
AnswerThe area of metal sheet required to make this hollow closed cone would be equal to the total surface area of the cone.
The formula of the total surface area of a cone with base radius $‘r’$ and slant height $‘l’$ is given as
Total Surface Area $=\pi\text{r}(\text{l}+\text{r})$ It is
given that the vertical height $‘h’ = 24cm$ and base radius $‘r’ = 7cm.$
To find the slant height ‘l’ we use the following relation Slant height, $\text{l}=\sqrt{\text{r}^2+\text{h}^2}$$\text{l}=\sqrt{7^2+24^2}$
$=\sqrt{49+576}$
$=\sqrt{625}$
$\text{l}=25$
Hence the slant height of the given cone is $25m$.
Now, substituting the values of $r = 7\ m$ and slant height $l = 25\ m$ and
using $\pi=\frac{22}{7}$ in the specified formula,
Total Surface Area $=\frac{(22).(7).(7+25)}{7} = (22)(32) = 704$
Therefore the total area of the metal sheet required to make the closed hollow cone is equal to $704\ cm^2.$
View full question & answer→Question 44 Marks
A bus stop is barricaded from the remaining part of the road, by using $50$ hollow cones made of recycled card board. Each cone has a base diameter of $40\ cm$ and height $1$ $m$ . If the outer side of each of the cones is to be painted and the cost of painting is $Rs$. $12$ per $m ^2$, what will be the cost of painting all these cones? (Use $\pi=3.14$ and $\sqrt{1.04}=1.02$ )
AnswerThe area to be painted is the curved surface area of each cone.
The formula of the curved surface area of a cone with base radius and slant height $7$ is given as:
Curved Surface Area $=\pi\text{rl}$ For each cone, we're given that the base diameter is $0.40\ m$
Hence the base radius $(r) = 0.20\ m$
The vertical height $(l) = 1\ m$ To find the slant height 'l' to be used in the formula for Curved Surface
Area we use the following relation Slant height,$\text{l}=\sqrt{\text{r}^2+\text{h}^2}$
$\text{l}=\sqrt{\text{0.2}^2+\text{1}^2}$
$\text{l}=\sqrt{\text{0.04}+\text{1}}$
$\text{l}=\sqrt{\text{1.04}}$
$\text{l}=1.02\text{m}$
Now substituting the values of $r =0.2 m$ and slant height $l =1.02 m$ and using $pi =3.14$ in the formula of
$C.S.A.$ We get Curved Surface Area $=(3.14)(0.2)(1.02)=0.64056 m^2$
This is the curved surface area of a single cone.
Since we need to paint $50$ such cones the total area to be painted is,
Total area to be painted $=(0.64056)(50)=32.028 m^2$
The cost of painting is given as $Rs.\ 12$ per $m ^2$
Hence the total cost of painting $=(12)(32.028)=384.336$
Hence, the total cost that would be incurred in the painting is $Rs.\ 384.336$.
View full question & answer→Question 54 Marks
Find the length of cloth used in making a conical pandal of height $100\ m$ and base radius $240\ m$, if the cloth is $100\pi\text{m}$ wide.
AnswerThe area of cloth required to make the conical pandal would be equal to the curved surface area of the cone.
The formula of the curved surface area of a cone with base radius $‘r’$ and slant height $‘l’$ is given as Curved Surface Area $=\pi\text{rl}$ It is given that the vertical height $‘h’ = 100m$ and base radius $‘r’ = 240m.$
To find the slant height ‘l’ we use the following relation Slant height, $\text{l}=\sqrt{\text{r}^2+\text{h}^2}$ $\text{l}=\sqrt{\text{240}^2+\text{100}^2}$ $\text{l}=\sqrt{\text{57600}+\text{10000}}$ $\text{l}=\sqrt{\text{67600}}$ $\text{l}=260$
Hence the slant height of the given cone is $260m$.
Now, substituting the values of $r = 240\ m$ and slant height $l = 260\ m$ in the formula for $C.S.A$,
We get Curved Surface Area $=(\pi)(240)(260)$
$=62400\pi$
Hence the area of the cloth required to make the conical pandal would be $62400\ \pi\text{m}^2$
It is given that the cloth is $100\pi$ wide.
Now, we can find the length of the cloth required by using the formula, Length of the canvas required $=\frac{\text{Area of the cloth}}{\text{Width of the cloth}}$$=\frac{62400\pi}{100\pi}$
$=624$
Hence the length of the cloth that is required is $624\ m$.
View full question & answer→Question 64 Marks
Find the volume of a conical tank with the following dimensions in liters: Radius $7\ cm$, slant height $25\ cm$.
AnswerIt is given that Radius of the cone $(r) = 7\ cm$ Slant height of the cone $(l) = 25\ cm$
As we know that,$ l^2 = r^2 + h^2$
$\text{h}=\sqrt{\text{l}^2-\text{r}^2}$
$\text{h}=\sqrt{\text{25}^2-\text{7}^2}$
$\text{h}=\sqrt{625-49}$
$=24\text{cm}$
Volume of a right circular cone$=\frac{1}{3}\pi\text{r}^2\text{h}$
$=\frac{1}{3}\times3.14\times7^2\times24$
$=1232\text{cm}^3=1.232\ \text{liters}$ $[1cm^3 = 0.01l]$
View full question & answer→Question 74 Marks
Two cones have their heights in the ratio $1 : 3$ and the radii of their bases in the ratio $3 : 1$. Find the ratio of their volumes.
AnswerLet ratio of the height of the cone be $h_x$ Height of the $1^{\text {st }}$
cone $=h_x$ Height of the of the $2^{\text {nd }}$ cone $=3 h_x$
Let the ratio of the radius of the of the cone $=r_x$
Radius of the $1^{\text {st }}$ cone $=3 r_x$
Radius of the $2^{\text {nd }}$ cone $=r_x$
The ratio of the volume $=v_1 / v_2$
Where $v_1=$ volume of $1^{\text {st }}$
cone $v_2=$ volume of $2^{\text {nd }}$ cone
$\Rightarrow\frac{\text{v}_1}{\text{v}_2}=\frac{\frac{1}{3}\pi\text{r}_\text{a}^2\text{h}_\text{a}}{\frac{1}{3}\pi\text{r}_\text{b}^2\text{h}_\text{b}}$
$=\frac{\text{r}_\text{a}^2\text{h}_\text{a}}{\text{r}_\text{b}^2\text{h}_\text{b}}$
$=\frac{3\text{r}_\text{x}^2\text{h}_\text{x}}{\text{r}_\text{x}^2\text{h}_\text{x}}$
$=\frac{9\text{r}_\text{x}^2\text{h}_\text{x}}{3\text{r}_\text{x}^2\text{h}_\text{x}}$
$=\frac{3}{1}$
$\Rightarrow\frac{\text{v}_1}{\text{v}_2}=\frac{3}{1}$
View full question & answer→Question 84 Marks
The ratio of volumes of two cones is $4 : 5$ and the ratio of the radii of their bases is $2 : 3$. Find the ratio of their vertical heights.
AnswerLet the ratio of the radius be $x$ Radius of $1^{\text {st }}$ cone $=2 x$
Radius of $2^{\text {nd }}$ cone $=3 x$ Let the ratio of the volume be $y$ Volume of $1^{\text {st }}$ cone $=4 y$ Volume of $2^{\text {nd }}$ cone $=5 y \frac{y_1}{y_2}=\frac{4 y}{5 y}$
$=\frac{4}{5}$
$\Rightarrow\frac{\frac{1}{3}\pi\text{r}_\text{a}^2\text{h}_\text{a}}{\frac{1}{3}\pi\text{r}_\text{b}^2\text{h}_\text{b}}=\frac{4}{5}$
$\Rightarrow\frac{\text{h}_\text{a}\times(\text{2x})^2}{\text{h}_\text{b}\times(\text{3x})^2}=\frac{4}5{}$
$\Rightarrow\frac{\text{h}_\text{a}\times\text{4x}^2}{\text{h}_\text{b}\times\text{9x}^2}=\frac{4}5{}$
$\Rightarrow\frac{\text{h}_\text{a}}{\text{h}_\text{b}}=\frac{36}{20}=\frac{9}{5}$
Therefore the heights are in the ratio of $9 : 5.$
View full question & answer→Question 94 Marks
Find the volume of a conical tank with the following dimensions in liters: Height $12\ cm$, slant height $13\ cm$.
AnswerIt is given that:
Height of the cone $(h) = 12\ cm$
Slant height of the cone $(l) = 13\ cm$
As we know that, $l^2 = r^2 + h^2$
$\text{r}=\sqrt{\text{l}^2-\text{h}^2}$
$\text{r}=\sqrt{\text{13}^2-\text{12}^2}$
$\text{r}=\sqrt{169-144}=\sqrt{25}$
$\text{r}=5\text{cm}$
Volume of a right circular cone:$=\frac{1}{3}\pi\text{r}^2\text{h}$
$=\frac{1}{3}\times3.14\times5^2\times10=314.85\text{cm}^2$
$=0.307\ \text{liters}.$ $[1cm^3 = 0.01]$
View full question & answer→Question 104 Marks
A right angled triangles of which the sides containing the right angle are $6.3\ cm$ and $10\ cm$ in length, is made to turn round on the longer side. Find the volume of the solid, thus generated. Also, find its curved surface area.
AnswerIt is given that:
Radius of cone$(r) = 6.3\ cm$
Height of the cone $(h) = 10\ cm$
We know thatSlant height $(l)$ $=\text{l}=\sqrt{\text{r}^2+\text{h}^2}$
$=\text{l}=\sqrt{\text{6.3}^2+\text{10}^2}=11.819\text{cm}$
Therefore Volume of cone $(v)$ $=\frac{1}{3}\pi\text{r}^2\text{h}$$=\frac{1}{3}\times3.14\times6.3^2\times10=415.8\text{cm}^3$
Curved surface area of cone $=\pi\text{rl}$ $=3.14\times6.3\times11.819=234.01\text{cm}^2$
View full question & answer→Question 114 Marks
What length of tarpaulin $4\ m$ wide will be required to make a conical tent of height $8\ m$ and base radius $6\ m$? Assume that the extra length of material will be required for stitching margins and wastage in cutting is approximately $20\ cm$. $($Use $\pi=3.14)$
AnswerGiven that,
Height of conical tent $(h)=8 m$
Radius of base of tent $( r )=6 m$
Slant height $( l )$
$\text { (l) }=\sqrt{r^2+h^2}$
$=\sqrt{8^2+6^2}$
$=\sqrt{100}=\sqrt{10} m$
$C.S.A$ of conical tent $=\pi rl =(3.14 \times 6 \times 10) m ^2=188.4 m^2$
Let the length of tarpaulin sheet required be $I$ As $20\ cm$ will wasted,
so effective Length will be $(1-0.2 m)$ Breadth of tarpaulin $=3 m$
Area of sheet $=$ $C.S.A$ of sheet $[I \times 0.2 \times 3] m =$ $188.4 m^2=1-0.2 m=62.8 m$
Accounting extra for wastage:
$\Rightarrow I=63 m$ Thus the length of the tarpaulin sheet will be $=63 m$
View full question & answer→Question 124 Marks
If the radius and slant height of a cone are in the ratio $7 : 13$ and its curved surface area is $286cm^2,$ find its radius.
AnswerIt is given that the curved surface area $(C.S.A)$ of the cone is $286cm^2$ and that the ratio between the base radius and the slant height is $7 : 13$.
The formula of the curved surface area of a cone with base radius $‘r’$ and slant height $‘l’$ is given as:
Curved Surface Area $=\pi\text{rl}$ Since only the ratio between the base radius and the slant height is given,
we shall use them by introducing a constant $‘k’$ So, $r = 7k l = 13k$
Substituting the values of $C.S.A$, base radius,
slant height and using $\pi=\frac{22}{7}$ in the above equation,
Curved Surface Area, $286=\frac{(22).(7\text{k}).(13\text{k})}{7}$
$286 = 286k^2 1 = k^2$
Hence the value of $k = 1$ From this we can find the value of base radius, $r = 7k r = 7$
Therefore the base radius of the cone is $7cm$.
View full question & answer→Question 134 Marks
The height of a conical vessel is $3.5\ cm$. If its capacity is $3.3$ litres of milk. Find its diameter of its base.
AnswerThe formula of the volume of a cone with base radius $‘r’$ and
vertical height $‘h’$ is given as: Volume of cone $=\frac{1}{3}\pi\text{r}^2\text{h}$
It is given that the height of the cone is $‘h’$ $= 3.5\ cm$ and that the volume of the cone is $3.3$ liters We know that,
$1$ liter $= 1000$ cubic centimeter Hence, the volume of the cone in cubic centimeter is $3300\text{cm}^3.$
We can now find the radius of base $‘r’$ by using the formula for the volume of a cone, while using
$\pi=\frac{22}{7}$$\text{r}^2=\frac{3(\text{Volume of the cone)}}{\pi\text{h}}$
$=\frac{(3)(3300)(7)}{(22)(3.5)}$
$\text{r}^2=900$
$\text{r}=30$
Hence the radius of the base of the cone with given dimensions is $‘r’ = 30cm.$
The diameter of base is twice the radius of the base. Hence the diameter of the base of the cone is $60cm.$
View full question & answer→Question 144 Marks
The radius and height of a right circular cone are in the ratio$ 5 : 12$ and its volume is $2512$ cubic cm. Find the slant height and radius of the cone. $($Use $\pi=3.14).$
AnswerLet the ratio be $y$ The radius of the cone $(r) = 5y$
Height of the cone $= 12y$ Now we know,
Slant height $(\text{l})=\sqrt{\text{r}^2+\text{h}^2}$
$=\sqrt{\text{5y}^2+\text{12y}^2}$
$=13\text{y}$
Now the volume of the cone is given $2512cm^3$
$\Rightarrow\frac{1}{3}\pi\text{r}^2\text{h}=2512$
$\Rightarrow\frac{1}{3}\times3.14\times\text{5y}^2\times\text{12y}=2512$
$\Rightarrow\text{y}^3=\frac{2512\times3}{3.14\times25\times2}$
$\Rightarrow\text{y}=2$
Therefore, Slant height $(l)$ $= 13y = 13 \times 2 = 26\ cm$
Radius of cone $= 5y = 5 \times 2 = 10\ cm$
View full question & answer→Question 154 Marks
The radius and the height of a right circular cone are in the ratio $5 : 12$. If its volume is $314$ cubic meter, find the slant height and the radius. $($Use $\pi=3.14).$
AnswerLet us assume the ratio to be $y$ Radius $(r) = 5y$ Height $(h) = 12y$
We know that: $l^2= r^2 + h^2 = 5y^2 + 12y^2 = 25y^2 + 144y^2 = 169^2 = 13y$
Now it is given that volume $= 314m^3$
$\Rightarrow\frac{1}{3}\pi\text{r}^2\text{h}=314\text{m}^3$
$\Rightarrow\frac{1}{3}\times3.14\times25\text{y}^2\times\text{12y}=314\text{m}^3$
$\Rightarrow\text{y}^3=1$
$\Rightarrow\text{y}=1$
Therefore, Slant height (l) $= 13y = 13m$ Radius $= 5y = 5m$
View full question & answer→Question 164 Marks
The circumference of the base of a $10m$ height conical tent is $44m$, calculate the length of canvas used in making the tent if width of canvas is $2m$ $\Big($Use $\pi=\frac{22}{7}\Big).$
AnswerWe know that $C.S.A$ of cone $=\pi\text{rl}$ Given circumference
$=2\pi\text{r}$
$\Rightarrow2\times\frac{22}{7}\times\text{r}=44$
$\Rightarrow\frac{\text{r}}{7}=1$
$\Rightarrow\text{r}=7\text{m}$
Therefore $\text{l}=\sqrt{\text{r}^2+\text{h}^2}$
$\text{l}=\sqrt{\text{7}^2+\text{10}^2}$
$\text{l}=\sqrt{149}\text{m}$
Therefore $C.S.A$ of tent $=\pi\text{rl}$$=\frac{22}{7}\times7\times\sqrt{149}$
$=22\sqrt{149}$
Therefore the length of canvas used in making the tent$=\frac{\text{Area of canvas}}{\text{Width of canvas}}$
$=\frac{22}{2\sqrt{149}}$
$=\frac{11}{\sqrt{149}}$
$=134.2\text{m}$
$=\frac{22}{7}\times24\times26$
$=\frac{1378}{7}\text{m}^2$
Cost of $1m^2$ canvas $= Rs\ 70$ Cost of $\frac{1378}{7}\text{m}^2$ canvas $\text{Rs.}=\frac{1378}{7}\times70$
$\text{Rs.}=1,37,280$ Thus the cost of canvas required to make the tent is $Rs.\ 1,37,280$.
View full question & answer→Question 174 Marks
A cylinder and a cone have equal radii of their base and equal heights. If their curved surface area are in the ratio $8 : 5$, show that the radius of each is to the height of each as $3 : 4$
AnswerIt is given that the base radius and the height of the cone and the cylinder are the same.
So let the base radius of each is $'r'$ and the vertical height of each is $'h'.$
Let the slant height of the cone be $'l'.$
The curved surface area of the cone $=\pi\text{rl}$
The curved surface area of the cylinder $=2\pi\text{rh}$
It is said that the ratio of the curved surface areas of the cylinder to that of the cone is $8 : 5$
So,$\frac{2\pi\text{rh}}{\pi\text{rl}}=\frac{8}{5}$
$\frac{\text{2h}}{\text{l}}=\frac{8}{5}$
$\frac{\text{h}}{\text{l}}=\frac{4}{5}$
But we know that $\text{l}=\sqrt{\text{r}^2+\text{h}^2}$
$=\frac{\text{h}^2}{\sqrt{\text{r}^2+\text{h}^2}}=\frac{4}{5}$
Squaring on both sides we get:$\frac{\text{h}^2}{\text{r}^2+\text{h}^2}=\frac{16}{25}$
$\frac{\text{r}^2}{\text{h}^2+\text{1}}=\frac{25}{16}$
$\frac{\text{r}^2}{\text{h}^2}=\frac{25}{16}-1$
$\frac{\text{r}^2}{\text{h}^2}=\frac{9}{16}$
$\frac{\text{r}}{\text{h}}=\frac{3}{4}$
Hence it is shown that the ratio of the radius to the height of the cone as well as the cylinder is: $3 : 4$.
View full question & answer→