Solution

\[ \begin{align*} \text{Height} (h) &= 15 \, \text{cm} \\ \text{Radius} (r) &= 7 \, \text{cm} \\ \\ \color{green} C.S.A &= \color{green}2\pi rh \\[6pt] &= 2 \times \frac{22}{\cancel{7}_{1}} \times \cancel{7}^{1} \times 15 \\[6pt] &= 44 \times 15 \\ &= 660 \, \text{cm}^2 \\ \\ \color{green} T.S.A &= \color{green} 2\pi r(h + r) \\[6pt] &= 2 \times \frac{22}{\cancel{7}_{1}} \times \cancel{7}^{1} \times (15 + 7) \\[6pt] &= 44 \times 22 \\[6pt] &= 968 \, \text{cm}^2 \end{align*} \]

Answer The curved surface area of the cylinder \( = \color{red} 660 \, \text{cm}^2 \) and the total surface area \( = \color{red} 968 \, \text{cm}^2 \).

Solution

\[ \begin{align*} \text{Diameter} &= 4.2 \, \text{dm} \\[6pt] \text{Radius} (r) &= \frac{4.2}{2} \implies 2.1 \, \text{dm} \\[6pt] \text{Height} (h) &= 1 \, \text{dm} \\ \\ (\text{Since } 1 \, \text{dm} &= 10 \, \text{cm}) \\ r &= 21 \, \text{cm} \\ h &= 10 \, \text{cm} \\ \\ \color{green} C.S.A &= \color{green} 2\pi rh \\[6pt] &= 2 \times \frac{22}{\cancel{7}_{1}} \times \cancel{21}^{3} \times 10 \\[6pt] &= 2 \times 22 \times 30 \\ &= 44 \times 30 \\ C.S.A &= 1320 \, \text{cm}^2 \end{align*} \]

Answer The area of the curved surface of the cylinder \( = \color{red} 1320 \, \text{cm}^2 \).

Solution

\[ \begin{align*} \text{Radius} (r) &= 10.5 \, \text{cm} \\ \text{Curved Surface Area} &= 1320 \, \text{cm}^2 \\ \color{green} C.S.A &= \color{green}2\pi rh \\[6pt] 2\pi rh & = 1320 \\[6pt] 2 \times \frac{22}{7} \times 10.5 \times h & = 1320 \\[6pt] \frac{22}{\cancel7_1} \times \cancel{21}^3 \times h & = 1320 \\[6pt] 66 \times h & = 1320 \\[6pt] h &= \frac{1320}{66} \\[6pt] h &= 20 \, \text{cm} \end{align*} \]

Answer The height of the cylinder \( = \color{red} 20 \, \text{cm} \).

Solution

\[ \begin{align*} \text{Circumference} &= 176 \ cm \\ \implies 2\pi r &= 176 \ cm \\ \text{Height }(h) &= 65 \ cm \\ \\ \color{green}\text{L.S.A} &= \color{green}2\pi r h \\[6pt] &= 2 \times \frac{22}{\cancel{7}_{1}} \times \cancel{28}^{4} \times 65 \\[6pt] &= 44 \times 4 \times 65 \\ &= 176 \times 65 \\ &= 11440 \ cm^2 \end{align*} \]

Answer The lateral surface area of the cylinder \(= \color{red}11440 \ cm^2 \)

Solution

Let the radii of the two cylinders be 2x and 3x and their respective heights 5y and 3y.

\[ \begin{align*} \text{CSA}_1 : \text{CSA}_2 &= \frac{\text{CSA}_1}{\text{CSA}_2} \\[6pt] &= \frac{2\pi \ r_1 \ h_1}{2\pi \ r_2 \ h_2} \\[6pt] &= \frac{\cancel{2\pi} \times 2 \cancel x \times 5 \cancel y}{\cancel{2\pi} \times 3 \cancel x \times 3 \cancel y} \\[6pt] &= \frac{10}{9} \\[6pt] \text{CSA}_1 : \text{CSA}_2 &= 10 : 9 \end{align*} \]

Answer The ratio of their curved surface areas is \(= \color{red} 10:9 \)

Solution

\[ \begin{aligned} \text{Diameter} &= 20 \,\text{cm} \\[6pt] r &= \frac{20}{2} \implies 10 \,\text{cm}\\[6pt] \text{Height}(h) &= 70 \,\text{cm}\\[6pt] \color{green}\text{C.S.A} &= \color{green}2\pi r h \\[6pt] & = 2 \times \frac{22}{\cancel7_1} \times 10 \times \cancel{70}^{10} \\[6pt] & = 44 \times 100 \\ &= 4400 \,\text{cm}^2\\[6pt] \text{Total cost} &= 4400 \times 4 \\ &= \text{₹} 17600 \end{aligned} \]

Answer Cost of painting \(= \color{red} \text{₹} 17600 \)

Solution

\[ \begin{aligned} r &= 10 \ cm\\ h &= 14 \ cm\\ \color{green}\text{T.S.A} &= \color{green} CSA + Base \ Area \\ &= \color{green} 2\pi rh + \pi r^2 \\ &= \pi r(2h + r) \\ & = 3.14 \times 10 \times (2 \times 14 + 10) \\ & = 31.4 \times (28 + 10) \\ & = 31.4 \times 38 \\ &= 1193.2 \ cm^2 \end{aligned} \]

Answer Total surface area \(= \color{red}1193.2 \ cm^2 \)

Solution

\[ \begin{aligned} \text{Diameter} (d) &= 21 \,\text{cm} \\[6pt] \text{Radius} (r) &= \frac{21}{2} \,\text{cm} \\[6pt] \text{Height }(h) &= 40 \,\text{cm}\\[6pt] \color{green}\text{C.S.A} &= \color{green}2\pi r h \\[6pt] & = \cancel 2 \times \frac{22}{\cancel7_1} \times \frac{\cancel{21}^3}{\cancel2} \times 40 \\[6pt] & = 66 \times 40 \\[6pt] \text{C.S.A} & = 2640 \ cm^2 \\[6pt] Revolutions &= 300 \\ & = 300 \times 2640 \\ \text{Area of room} & = 792000 \ cm^2 \\[6pt] 1 \ cm^2 & = \frac{1}{10000} \ m^2 \\[6pt] & = \frac{792000}{10000} \ m^2 \\[6pt] &= 79.2\,\text{m}^2 \end{aligned} \]

Answer The area of the room is \( \color{red}79.2 \ m^2 \)

Solution

\[ \begin{aligned} \text{Diameter} &= 56 \,\text{cm} \\ r &= \frac{56}{2} \\ & = 28 \,\text{cm} \Rightarrow 0.28 \,\text{m} \\[6pt] h &= 2.25 \,\text{m} \\[6pt] \color{green} \text{T.S.A} &= \color{green} 2\pi r(h + r) \\[6pt] &= 2 \times \frac{22}{\cancel7_1} \times \cancel{0.28}^{.04} \times (2.25 + 0.28) \\[6pt] &= 44 \times 0.04 \times 2.53 \\[6pt] &= 1.76 \times 2.53 \\[6pt] &= 4.4528 \,\text{m}^2 \\[6pt] \text{Total Cost} &= 4.4528 \times 80 \\[6pt] &= \text{₹} 356.22 \end{aligned} \]

Answer The cost of the metal used to make the cylinder is \( \color{red} \text{₹} \, 356.22 \)

Solution

\[ \begin{aligned} \text{Circumference} \ (2\pi r) &= 22 \ cm \\ h &= 10 \ cm \\ \color{green}\text{Surface Area} &= \color{green}2\pi r h \\[6pt] &= 22 \times 10 \\ &= 220 \ cm^2 \end{aligned} \]

Answer The surface area of the cylinder is \( \color{red}220 \ cm^2 \)

Solution

\[ \begin{aligned} \text{Outer diameter} &= 21 \, \text{m} \\[6pt] R &= \frac{21}{2} \Rightarrow 10.5 \, \text{m} \\[6pt] \text{Inner diameter} &= 14 \, \text{m} \\[6pt] r &= \frac{14}{2} \Rightarrow 7 \, \text{m} \\[6pt] \text{Height } (h)&= 21 \, \text{m} \\[6pt] \color{green}\text{Total Area} &= \color{green}\text{Outer C.S.A} + \text{Inner C.S.A} \\[6pt] &= 2\pi R h + 2\pi r h \\[6pt] &= 2\pi h (R + r ) \\[6pt] &= 2 \times \frac{22}{\cancel7_1} \times \cancel{21}^3 \times (10.5 + 7 ) \\[6pt] &= 44 \times 3 \times 17.5 \\[6pt] &= 132 \times 17.5 \\[6pt] &= 2310 \ m^2 \\[6pt] \text{Total Cost} &= 2310 \times 25 \\ & = \text{₹} 57750 \end{aligned} \]

Answer Total cost of renovation \( = \color{red} \text{₹} 57750 \)