**Solution:**

\begin{align*} \text{Curved surface area of cylinder} &= 2 \pi rh \\ or, 616 cm^2&= 2 \times \frac{22}{7} \times r \times 14 \: cm \\ or, 616cm^2 &= 88r \: cm \\ or, r&= \frac{616}{88}cm \\ \therefore radius (r) &= 7 \: cm \: _{Ans} \end{align*}

**Solution:**

Base radius (r) = 28 cm

Height (h) = 72 cm

Total surface area (T.S.A) = ?

We know that,

\begin{align*} T.S.A &= 2 \pi r (r+h)\\ &= 2\times \frac{22}{7} \times 28cm (28cm+72cm) \\ &= 8 \times 22cm (100cm)\\ &= 17600 \: sq. cm \: _{Ans} \end{align*}

**Solution:**

r = 14 cm, h = 13 cm

\begin{align*} Volume \: of \: cylinder &= \pi r^2 h \\ &= \frac{22}{7} \times (14cm)^2 \times 13cm \\ &= 8008cm^3 \: \: _{Ans}\end{align*}

**Solution:**

h = 60 cm

\(r=\frac{14}{2}=7cm\)

\begin{align*} \text{Volume of half part of a cylinder} &= \frac{1}{2} \times \pi r^2 h \\ &= \frac{1}{2} \times \frac{22}{7} \times 7^2 \times 60 \:cm^3 \\ &= 11\times 7 \times 60cm^3 \\ &= 4620 cm^3 \: \: _{Ans} \end{align*}

**Solution:**

Area of base \( (\pi r^2 ) = 154 cm^2 \)

height (h) = 14 cm

\begin{align*} Volume \: (v) &= \pi r^2 h \\ &= 154cm^2 \times 14 cm \\ &= 2156 \: cm^3 \: _{Ans} \end{align*}

**Solution:**

r + h = 10cm

\(Circumference = 2\pi r = 308 cm\)

\begin{align*}Total \: surface \: area &= 2 \pi r (r+h) \\ &= 308 \times 10 \\ &= 3080 cm^2 \: _{ans} \end{align*}

**Solution:**

\(r + h = 34 cm \\ Total \: surface \: area \: (S) = 2992cm^2 \\ By\: formula, \)

\begin{align*} S &= 2 \pi r (r+h)\\ or, 2992 &= 2 \times \frac{22}{7} \times r(34)\\ or, 2992 \times 7&= 1496r\\or, r &= \frac{2992\times 7}{1496}\\ \therefore r &= 14 \: cm \: _{Ans}\end{align*}

**Solution:**

Here,

\begin{align*} \text{surface area of cylindrical wood} &= 2 \pi rh \\ or, 308 &= 2 \pi h^2 \: [\because r = h ] \\ or, h^2 &= \frac{308}{2 \pi }\\ or, h^2 &= \frac{308}{2} \times \frac{22}{7}\\ or, h^2 &= \frac{308}{44} \times 7 \\ or, h &= \sqrt{49}\\ \therefore h &= 7 \: cm \: \: _{Ans}\end{align*}

Volume of the cylinder (V) = \(\pi\)r^{2}h

or, 1078 = \(\frac {22}7\)× r^{2}× 7

or, 1078 = 22r^{2}

or, r^{2} = \(\frac {1078}{22}\)

or, r^{2} = 49

or, r = \(\sqrt {49}\)

∴ r = 7 cm_{Ans}

Volume of cylinder (V) = \(\pi\)r^{2}h = area of base \(\times\) height

or, 1540 = 154 \(\times\) height

or, height = \(\frac {1540}{154}\)

∴ height of the solid = 10 cm_{Ans}

Radius of the base (r) = \(\frac {1.4}2\) = 0.7 m

height of the solid = ?

Volume of the solid (V) = 770 liters = \(\frac {770}{1000}\)m^{3} = 0.77 m^{3}

Nw,

Volume = \(\pi\)r^{2}h

or, 0.77 = \(\frac {22}{7}\) \(\times\) (0.7)^{2} \(\times\) h

or, 0.77 = \(\frac {22}7\) \(\times\) 0.49h

or, 0.77 = 22 \(\times\) 0.07h

or, 0.77 = 1.54 h

or, h = \(\frac {0.77}{1.54}\)

∴ height of the solid (h) = 0.5 m_{Ans}

####
The volume of a cylindrical can is 1.54 litre. If the area of its base is 77 m^{3}, find its height.

^{3}, find its height.

Volume (V) = 1.54 liters = 1.54 \(\times\) 1000 cm^{3} = 1540 cm^{3}

Area of base (A) = 77 cm^{2}

Height (h)= ?

Here,

V = A \(\times\) h

or, 1540 = 77 \(\times\) h

or, h = \(\frac {1540}{77}\)

∴ height of the can (h) = 20 cm_{Ans}

Here,

r = 6 cm

By Question,

curved surface area = \(\frac 23\) \(\times\) total surface area

or, 2\(\pi\)rh = \(\frac 23\) \(\times\) 2\(\pi\)r (r + h)

or, h = \(\frac 23\) (r + h)

or, 3h = 2(6 + h)

or, 3h = 12 + 2h

or, 3h - 2h = 12

∴ height of the cylinder (h) = 12 cm_{Ans}

Here,

h = 25 cm

r_{1} = 4 cm

t = 1 cm

r_{2} = (4 - 1)cm = 3 cm

Now,

\begin{align*} \text {Volume of the metal (V)} &= \pi(r_1^2 - r_2^2)h\\ &=\frac {22}7(4^2 - 3^2) \times 25\\ &= \frac {22}{7} \times 7 \times 25\\ &= 550 cm^3_{Ans}\\ \end{align*}

Base area of a cylinder (A) = 154 cm^{2}

Curved Surface Area (CSA) = 880 cm^{2}

Total Surface Area (TSA) = ?

By formula,

\begin{align*} TSA &= 2A + CSA\\ &= (2 \times 154 cm^2) + 880 cm^2\\ &= (308 + 880)cm^2\\ &= 1188cm^2_{Ans}\\ \end{align*}