Surface Area of a Combination of Solids Questions in English

(D) Given, radius $r = 4\, cm$ and height $h = 3\, cm$.
First, we calculate the slant height $l$ using the formula $l = \sqrt{r^{2} + h^{2}}$.
$l = \sqrt{4^{2} + 3^{2}} = \sqrt{16 + 9} = \sqrt{25} = 5\, cm$.
The Total Surface Area $(TSA)$ of a cone is given by the formula $TSA = \pi r(r + l)$.
Substituting the values, $TSA = \pi \times 4 \times (4 + 5) = \pi \times 4 \times 9 = 36\pi\, cm^{2}$.

(A) For the given cone,the radius $r = 5\,cm$ and the height $h = 12\,cm$.
First,we calculate the slant height $l$ using the formula $l = \sqrt{r^2 + h^2}$.
$l = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13\,cm$.
The $CSA$ (Curved Surface Area) of a cone is given by $\pi r l$.
The area of the base of the cone is $\pi r^2$.
Therefore,the ratio of the $CSA$ to the area of the base is $\frac{\pi r l}{\pi r^2} = \frac{l}{r}$.
Substituting the values,we get $\frac{13}{5}$,which is $13:5$.

(D) The formula for the surface area of a sphere is $A = 4 \pi r^{2}$.
Given the radius $r = 7\,cm$.
Substituting the value of $r$ into the formula:
$A = 4 \times \frac{22}{7} \times (7)^{2}$
$A = 4 \times \frac{22}{7} \times 49$
$A = 4 \times 22 \times 7$
$A = 88 \times 7 = 616\,cm^{2}$.

(B) The diameter of the hemisphere is $d = 20 \, cm$.
Therefore, the radius $r = d / 2 = 20 / 2 = 10 \, cm$.
The formula for the Curved Surface Area $(CSA)$ of a hemisphere is $2 \pi r^{2}$.
Substituting the value of $r$:
$CSA = 2 \times \pi \times (10)^{2}$
$CSA = 2 \times \pi \times 100$
$CSA = 200 \pi \, cm^{2}$.

(C) The formula for the Total Surface Area $(TSA)$ of a hemisphere is $3 \pi r^{2}$.
Given the radius $r = 10 \, cm$.
Substituting the value of $r$ in the formula:
$TSA = 3 \times \pi \times (10)^{2}$
$TSA = 3 \times \pi \times 100$
$TSA = 300 \pi \, cm^{2}$.

(B) The diameter of the hemisphere is $d = 10 \, cm$.
Therefore, the radius $r = d / 2 = 10 / 2 = 5 \, cm$.
The formula for the $TSA$ of a hemisphere is $3 \pi r^2$.
Substituting the value of $r$ into the formula:
$TSA = 3 \times \pi \times (5)^2$
$TSA = 3 \times \pi \times 25$
$TSA = 75 \pi \, cm^2$.

(D) The area of the circular base of a hemisphere is given by the formula $\pi r^{2}$.
Given,$\pi r^{2} = 154 \, cm^{2}$.
The total surface area $(TSA)$ of a hemisphere is the sum of the curved surface area $(2\pi r^{2})$ and the base area $(\pi r^{2})$,which equals $3\pi r^{2}$.
Therefore,$TSA = 3 \times (\pi r^{2}) = 3 \times 154 = 462 \, cm^{2}$.

(D) Let the radii of the two spheres be $r_{1}$ and $r_{2}$ units.
Given that the ratio of their radii is $\frac{r_{1}}{r_{2}} = \frac{3}{2}$.
The surface area of a sphere is given by the formula $A = 4\pi r^{2}$.
The ratio of the surface areas of the two spheres is $\frac{A_{1}}{A_{2}} = \frac{4\pi r_{1}^{2}}{4\pi r_{2}^{2}}$.
Simplifying this,we get $\frac{A_{1}}{A_{2}} = \left(\frac{r_{1}}{r_{2}}\right)^{2}$.
Substituting the given ratio: $\frac{A_{1}}{A_{2}} = \left(\frac{3}{2}\right)^{2} = \frac{9}{4}$.
Therefore,the ratio of their surface areas is $9:4$.

(D) Let the radius of both the sphere and the solid hemisphere be $r$.
The surface area of a sphere is given by the formula $4 \pi r^2$.
The total surface area $(TSA)$ of a solid hemisphere is given by the formula $3 \pi r^2$.
Therefore,the ratio of the surface area of the sphere to the $TSA$ of the hemisphere is $\frac{4 \pi r^2}{3 \pi r^2} = \frac{4}{3}$.
Thus,the required ratio is $4:3$.

(D) Given: Diameter of the cylindrical tank $= 2.1 \, m$.
Radius $r = \frac{\text{diameter}}{2} = \frac{2.1}{2} = 1.05 \, m$ or $\frac{21}{20} \, m$.
Height $h = 5 \, m$.
The formula for the Curved Surface Area $(CSA)$ of a cylinder is $2 \pi r h$.
Substituting the values: $CSA = 2 \times \frac{22}{7} \times \frac{21}{20} \times 5$.
$CSA = 2 \times \frac{22}{7} \times \frac{21}{20} \times 5 = 2 \times 22 \times \frac{3}{20} \times 5 = 44 \times \frac{15}{20} = 44 \times 0.75 = 33 \, m^2$.
Therefore,the $CSA$ is $33 \, m^2$.

(B) Given: Height $(h) = 3\,m$,Diameter $= 8\,m$.
Radius $(r) = \frac{\text{diameter}}{2} = \frac{8}{2} = 4\,m$.
For a cone,the slant height $(l)$ is given by $l = \sqrt{r^{2} + h^{2}}$.
$l = \sqrt{4^{2} + 3^{2}} = \sqrt{16 + 9} = \sqrt{25} = 5\,m$.
$CSA$ of the conical tank $= \pi r l$.
$CSA = 3.14 \times 4 \times 5 = 3.14 \times 20 = 62.8\,m^{2}$.

(A) The surface area of a sphere is given by the formula $A = 4 \pi r^{2}$.
Given $A = \frac{763}{3} \, m^{2}$ and $\pi = 3.14$.
Substituting the values into the formula:
$\frac{763}{3} = 4 \times 3.14 \times r^{2}$
$\frac{763}{3} = 12.56 \times r^{2}$
$r^{2} = \frac{763}{3 \times 12.56}$
$r^{2} = \frac{763}{37.68}$
$r^{2} \approx 20.25$
Taking the square root of both sides:
$r = \sqrt{20.25} = 4.5 \, m$.
Thus,the radius of the sphere is $4.5 \, m$.

(C) The formula for the surface area of a sphere is $A = 4\pi r^{2}$.
Given that the surface area $A = 1256\,cm^{2}$ and $\pi = 3.14$.
Substituting the values into the formula:
$1256 = 4 \times 3.14 \times r^{2}$
$1256 = 12.56 \times r^{2}$
$r^{2} = \frac{1256}{12.56}$
$r^{2} = 100$
Taking the square root on both sides,we get $r = 10\,cm$.

(C) For the frustum of a cone, the slant height $l$ is given by the formula:
$l = \sqrt{h^2 + (r_1 - r_2)^2}$
Given $h = 4\,cm$, $r_1 = 10\,cm$, and $r_2 = 7\,cm$:
$l = \sqrt{4^2 + (10 - 7)^2} = \sqrt{16 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5\,cm$
The curved surface area $(CSA)$ of a frustum of a cone is given by:
$CSA = \pi l(r_1 + r_2)$
Substituting the values:
$CSA = \pi \times 5 \times (10 + 7) = \pi \times 5 \times 17 = 85\pi\,cm^2$

(B) The Total Surface Area $(TSA)$ of a hemisphere is given by the formula: $TSA = 3\pi r^{2}$.
Given,$TSA = 462\,cm^{2}$.
So,$3\pi r^{2} = 462$.
Using $\pi = \frac{22}{7}$,we have $3 \times \frac{22}{7} \times r^{2} = 462$.
$r^{2} = 462 \times \frac{7}{3 \times 22}$.
$r^{2} = 462 \times \frac{7}{66}$.
$r^{2} = 7 \times 7 = 49$.
$r = \sqrt{49} = 7\,cm$.
The diameter $d = 2r = 2 \times 7 = 14\,cm$.

(D) The formula for the total surface area of a cube with side length $a$ is given by $6a^{2}$.
Given the side length $a = 4 \, cm$.
Substituting the value of $a$ into the formula:
Total Surface Area $= 6 \times (4 \, cm)^{2}$
Total Surface Area $= 6 \times 16 \, cm^{2}$
Total Surface Area $= 96 \, cm^{2}$.
Therefore,the correct option is $D$.

(A) The formula for the total surface area of a cuboid is $2(lb + bh + hl)$,where $l$,$b$,and $h$ are the length,breadth,and height of the cuboid respectively.
Given dimensions are $l = 15\,cm$,$b = 12\,cm$,and $h = 10\,cm$.
Substituting these values into the formula:
Total surface area $= 2(15 \times 12 + 12 \times 10 + 10 \times 15)$
$= 2(180 + 120 + 150)$
$= 2(450)$
$= 900\,cm^{2}$.

(B) For the given hemisphere,the radius $r = \frac{\text{diameter}}{2} = \frac{2}{2} = 1 \, cm$.
The total surface area of a hemisphere is given by the formula $3 \pi r^{2}$.
Substituting the value of $r = 1 \, cm$ into the formula:
Total surface area $= 3 \times \pi \times (1)^{2} = 3 \pi \, cm^{2}$.

(C) The formula for the curved surface area of a cylinder is $2 \pi r h$.
Given that the curved surface area is $132 \, cm^2$ and the height $h = 3 \, cm$.
Substituting the values into the formula:
$132 = 2 \times \frac{22}{7} \times r \times 3$
$132 = \frac{132}{7} \times r$
Multiplying both sides by $\frac{7}{132}$:
$r = 132 \times \frac{7}{132} = 7 \, cm$.
The diameter $d$ is twice the radius:
$d = 2r = 2 \times 7 = 14 \, cm$.

(D) The formula for the total surface area of a cylinder is $2 \pi r(h + r)$.
Given,radius $r = 7 \, cm$ and height $h = 13 \, cm$.
Substituting the values into the formula:
Total surface area $= 2 \times \frac{22}{7} \times 7 \times (13 + 7)$
$= 2 \times 22 \times 20$
$= 44 \times 20 = 880 \, cm^{2}$.

(C) For the given cone,the radius $r = \frac{\text{diameter}}{2} = \frac{10}{2} = 5 \, cm$.
The curved surface area $(CSA)$ of a cone is given by the formula $CSA = \pi r l$,where $l$ is the slant height.
Given that $CSA = 85 \pi \, cm^{2}$,we have:
$85 \pi = \pi \times 5 \times l$
Dividing both sides by $5 \pi$:
$l = \frac{85 \pi}{5 \pi} = 17 \, cm$.
Therefore,the slant height of the cone is $17 \, cm$.

(C) cylinder with cone-shaped lids at both ends consists of the curved surface area of the cylinder and the curved surface areas of the two cones.
$1$. The curved surface area of the cylinder is $2\pi rh$,where $r$ is the radius and $h$ is the height of the cylinder.
$2$. The curved surface area of one cone is $\pi rl$,where $l$ is the slant height of the cone.
$3$. Since there are two such cones,their combined curved surface area is $2\pi rl$.
$4$. The total surface area is the sum of these areas: $2\pi rh + 2\pi rl = 2\pi r(h+l)$.