Base are of the cone = \(\pi\)r^{2} = 125 cm^{2}

Height of the cone (h) = 9 cm

\begin{align*} Volume (V) &= \frac 13 {\pi}r^2h\\ &= \frac 13 × 125 × 9 cm^3\\ &= 125 × 3 cm^3\\ &= 375 cm^3_{Ans}\\ \end{align*}

Here,

\begin{align*} {\text{Radius of cone (r)}} &=\sqrt {l^2 - h^2}\\ &= \sqrt {(25 cm)^2 - (24 cm)^2}\\ &= \sqrt {625 cm^2 - 576 cm^2}\\ &= \sqrt {49 cm^2}\\ &= 7 cm\\ \end{align*}

Now,

\begin{align*} {\text{Volume of the cone (V)}} &=\frac 13 {\pi}r^2h\\ &= \frac 13 × \frac {22}7 × (7 cm)^2 × 24 cm\\ &= 22 × 7 × 8 cm^3\\ &= 1232 cm^3_{Ans}\\ \end{align*}

Here,

l = 50 cm

h = 48 cm

Then,

\begin{align*} r &= \sqrt {l^2 - h^2}\\ &= \sqrt {50^2 - 48^2} cm\\ &= \sqrt {196} cm\\ &= 14 cm\\ \end{align*}

\begin{align*} \therefore {\text{Volume of the cone}} &=\frac 13 {\pi}r^2h\\ &= \frac 13 × \frac {22}7 × 14^2 × 48 cm^3\\ &=\ \frac {22 × 196 × 48}{21} cm^3\\ &= 9856 cm^3_{Ans} \end{align*}

Here,

r = 14 cm,

V = 1848 cm^{3}

h = ?

By formula,

V = \(\frac 13\)\(\pi\)r^{2}h

or, 1848 = \(\frac 13\)× \(\frac {22}7\)× (14)^{2}× h

or, 1848× 21 = 22× 196 h

or, h = \(\frac {1848 × 21}{22 × 196}\) cm

∴ h = 9 cm

Hence, the height of cone is 9 cm._{Ans}

Here,

l = 25 cm

h = 24 cm

If r be the radius of base, then:

\begin{align*} r &= \sqrt {l^2 - h^2}\\ &= \sqrt {(25)^2 - (24)^2}\\ &= \sqrt {49}\\ &= 7 cm\\ \end{align*}

Now,

\begin{align*} {\text{Curved Surface Area}} &= {\pi}rl\\ &= \frac {22}7 × 7 × 25 cm^2\\ &= 550 cm^2_{Ans}\\ \end{align*}

Here,

r = 5 cm

h = 4 cm

Curved Surface Area of cone (S) = ?

Here,

\begin{align*} r &= \sqrt {l^2 - h^2}\\ &= \sqrt {5^2 - 4^2}\\ &= \sqrt {25 - 16}\\ &= \sqrt 9\\ &= 3 cm\\ \end{align*}

\begin{align*} S &= {\pi}rl\\ &= \frac {22}7× 3× 5 cm^2 \\ &= 47.14 cm^2_{Ans}\\ \end{align*}

Here,

2\(\pi\)r = 88 cm

or, \(\pi\) = \(\frac {88 × 7}{2 × 22} cm\)

or, r = 2× 7 cm

∴ r = 14 cm

l = 30 cm

Now,

\begin{align*} {\text{Curved Surface Area of the cone}} &= {\pi}rl\\ &= \frac {22}7× 14× 30 cm^2\\ &= 44× 30 cm^2\\ &= 1320 cm^2_{Ans}\\ \end{align*}

Here,

(l + r) = 32 cm

We have,

Circumference = 2\(\pi\)r

or, 44 cm = 2\(\pi\)r

or, \(\pi\)r = \(\frac {44}2\)

∴ \(\pi\)r = 22 cm

Now,

\begin{align*} {\text{Total Surface Area of Cone}} &= {\pi}r (l + r)\\ &= 22 cm × 32 cm\\ &= 704 cm^2_{Ans}\\ \end{align*}

Here,

r = 9 cm

l = 15 cm

Now,

\begin{align*} {\text{Total Surface Area}} & = {\pi}r^2 + {\pi}rl\\ &= {\pi}r (r + l)\\ &= \frac {22}7 × 9 (9 + 15) cm^2\\ &= \frac {22}7 × 9 × 24 cm^2\\ &= 678.857 cm^2_{Ans}\\ \end{align*}

Here,

l = 100 cm

Curved Surface Area = \(\pi\)rl

or, 8800 cm^{2} = \(\frac {22}7\)× r× 100 cm

or, r = \(\frac {8800 cm^2 × 7}{22 × 100 cm}\)

∴ r = 28 cm

Radius (OR) = 28 cm

In right angled \(\triangle\)POR,

\begin{align*} PO &= \sqrt {PR^2 - OR^2}\\ &= \sqrt {100^2 - 28^2}cm\\ &= 96 cm_{Ans}\\ \end{align*}

Here,

r + l = 32 cm

Toral surface area of the cone = \(\pi\)r (r + l)

By Question,

Total Surface Area = 4928 cm^{2}

or, \(\pi\)r (r + l) = 4928 cm^{2}

or, \(\frac {22r}7\) (32 cm) = 4928 cm^{2}

or, r = 4928× \(\frac {7 cm^2}{22 × 32 cm}\)

∴ r = 49 cm_{Ans}

**Note: H**ere, r + l = 32< 49 = r, which is impossible. So, the question is wrong.

Suppose,

r = 7x

h = 12x

Volume (V) = 616 cm^{3}

By formula,

V = 616

or, \(\frac 13\) \(\pi\)r^{2}h = 616

or, \(\frac 13\)× \(\frac {22}7\)× (7x)^{2} . 12x = 616

or, \(\frac {22 × 49x^2 × 12x}{21}\) = 616

or, 616x^{3} = 616

or, x^{3} = 1

∴ x = 1

∴ r = 7x = 7 cm

∴ h = 12x = 12 cm

Now,

\begin{align*} l &=\sqrt {r^2 + h^2}\\ &= \sqrt {7^2 + 12^2}\\ &= \sqrt {193}\\ \end{align*}

\begin{align*} {\text{Curved Surface Area (CA)}} &= {\pi}rl\\ &= \frac {22}7 × 7 × \sqrt {193}\\ &= 305.63 cm^2_{Ans}\\ \end{align*}

Here,

TSA = 704 cm^{2}

CA = 550 cm^{2}

By formula,

TSA = \(\pi\)r (r + l)

or, 704 = \(\pi\)r (r + l)

or, 704 = \(\pi\)r^{2} + \(\pi\)rl...........................(i)

Again,

CA = \(\pi\)rl

or, 550 = \(\pi\)rl...........................(ii)

From (i) and (ii)

704 = \(\pi\)r^{2} + 550

or, 154 = \(\pi\)r^{2}

or, r^{2} = \(\frac {154}{\pi}\)

or, r^{2} = \(\frac {154 × 7}{22}\)

or, r^{2} = 49

∴ r = 7 cm_{Ans}