text: Here,
  
  r = 15 cm
  
  \begin{align*} \text{Total Surface Area of hemisphere} &= 3{\pi}r^2\\ &= 3 \times \frac {22}7 \times 15 \times 15\\ &= \frac {14850}{7}\\ &= 2121.43 cm^2_{Ans}\\ \end{align*}
  
  Surface Area of the sphere (A) = 616 cm2
  
  radius of the sphere (r) = ?
  
  By formula,
  
  A = 4$$\pi$$r2
  
  or, r2 = $$\frac A{4\pi}$$
  
  or, r2 = $$\frac {616 \times 7}{4 \times 22}$$
  
  or, r2 = 49
  
  ∴ r = 7cmAns
  
  Suppose,
  
  Then,
  
  Surface Area of Sphere = 4$$\pi$$r2
  
  or, $$\pi$$ = 4$$\pi$$r2
  
  or, r2 = $$\frac {\pi}{4\pi}$$
  
  or, r2 = $$\frac 14$$
  
  or, r = $$\sqrt {\frac 14}$$
  
  ∴ r = $$\frac 12$$cmAns
  
  Here,
  
  r = $$\frac {42 cm}{2}$$ = 21 cm
  
  \begin{align*} \text{Volume of the hemisphere (V)} &=\frac 23{\pi}r^3\\ &= \frac 23 \times \frac {22}7 \times (21)^3\\ &= 44 \times (21)^2\\ &= 19404 cm^3_{Ans}\\ \end{align*}
  
  Here,
  
  Volume of marble (V) = $$\frac {\pi}6cm^3$$
  
  or, $$\frac 43$$ $$\pi$$r3 = $$\frac {\pi}6cm^3$$
  
  or, r3 = $$\frac {\pi}6$$ $$\times$$ $$\frac 3{4\pi}cm^3$$
  
  or, r3 = $$\frac 1{2^3}cm^3$$
  
  ∴ r = $$\frac 12cm$$
  
  Hence, the diameter of the marble = 2 $$\times$$ r = 2 $$\times$$ $$\frac 12$$ cm = 1 cmAns
  
  Let r be the radius of the sphere.
  
  Volume = $$\frac 43$$$$\pi$$r3
  
  or, 38808 = $$\frac 43$$ $$\times$$ $$\frac {22}7$$ $$\times$$ r3
  
  or, r3 = 38808 $$\times$$ $$\frac {21}{88}$$
  
  or, r3 = 441 $$\times$$ 21
  
  or, r3 = (21)3
  
  ∴ r = 21Ans
  
  Here,
  
  Volume of sphere = $$\frac {3773}{21}cm^3$$
  
  or, $$\frac 43$$$$\pi$$r3 = $$\frac {3773}{21}cm^3$$
  
  or, $$\frac 43$$ $$\times$$ $$\frac {22}7$$ $$\times$$ r3 = $$\frac {3773}{21}cm^3$$
  
  or, $$\frac {88}{21}r^3$$ = $$\frac {3773}{21}cm^3$$
  
  or, r3 = $$\frac {3773}{21}$$ $$\times$$ $$\frac {21}{88}cm^3$$
  
  or, r3 = $$\frac {343}{8}cm^3$$
  
  or, r3 = ($$\frac 73$$cm)3
  
  ∴ r = $$\frac 72$$cm
  
  Hence, Circumference of sphere = 2$$\pi$$r = 2 $$\times$$ $$\frac {22}7$$ $$\times$$ $$\frac 72$$ cm = 22 cmAns
  
  Here,
  
  Volume of spherical solid = $$\frac 43$$$${\pi}cm^3$$
  
  or, $$\frac 43$$$${\pi}r^3$$ = $$\frac 43$$$${\pi}cm^3$$
  
  or, r3 = $$\frac {\frac {4\pi}{3}}{\frac {4\pi}{3}}cm^3$$
  
  or, r3 = 1 cm3
  
  ∴ r = 1 cm
  
  Hence, the radius of spherical solid is 1 cm.Ans
  
  Here,
  
  Volume of sphere (V) = 36$${\pi}cm^3$$
  
  Diameter of a sphere (d) = ?
  
  We know that,
  
  V = $$\frac 16$$$${\pi}d^3$$
  
  or, d = $$\sqrt [3]\frac{6V}{d}$$
  
  or, d = $$\sqrt [3]\frac {6 \times 36\pi}{\pi}$$
  
  or, d = $$\sqrt [3]{6 \times 6 \times 6}$$
  
  ∴ d = 6cm
  
  \begin{align*} \text{The Surface Area (A)} &= {\pi}d^2\\ &= \frac {22}7 \times (6cm)^2\\ &= \frac {22}7 \times 36 cm^2\\ &= 113.14cm^2_{Ans}\\ \end{align*}
  
  Here,
  
  Volume (V) = $$\frac {1372\pi}{3}cm^3$$
  
  Total Surface Area (S) = ?
  
  By Formula,
  
  V = $$\frac 43$$$${\pi}r^3$$
  
  or, $$\frac {1372\pi}{3}$$ =$$\frac 43$$$${\pi}r^3$$
  
  or, r3 =$$\frac {1372\pi}{3}$$ $$\times$$ $$\frac 3{4\pi}$$
  
  or, r3 = 343
  
  or, r3 = 73
  
  ∴ r = 7
  
  We know that,
  
  \begin{align*} S &= 4{\pi}r^2\\ &= 4 \times \frac {22}7 \times 7^2 cm^2\\ &= 616 cm^2_{Ans}\\ \end{align*}
  
  Here,
  
  Surface Area of sphere = 616 cm2
  
  or, 4$${\pi}r^2$$ = 616 cm2
  
  or, $${\pi}r^2$$ = $$\frac {616}4 cm^2$$
  
  ∴$${\pi}r^2$$ = 154 cm2
  
  We know that,
  
  \begin{align*} \text{Total Surface Area of a hemisphere} &= 3{\pi}r^2\\ &= 3 \times 154 cm^2\\ &= 462 cm^2_{Ans}\\ \end{align*}
  
  Here,
  
  Total Surface Area of hemisphere (S) = 243 $${\pi}cm^2$$
  
  Volume (V) = ?
  
  We know that,
  
  S = 3$${\pi}r^2$$
  
  or, 243$$\pi$$ = 3$${\pi}r^2$$
  
  or, r2 = $$\frac {243\pi}{3\pi}$$
  
  or, r2 = 81
  
  ∴ r = 9 cm
  
  Now,
  
  \begin{align*} V &= \frac 23 {\pi}r^3\\ &= \frac 23 \times \frac {22}7 \times 9^3\\ &= 1527.43 cm^3_{Ans}\\ \end{align*}
  
  Here,
  
  4$${\pi}r^2$$ = $$\frac 1{4\pi}$$
  
  or, r2 = $$\frac 1{(4\pi)^2}$$
  
  ∴r = $$\frac 1{4\pi}$$
  
  Now,
  
  \begin{align*} \text{Volume (V)} &= \frac 43 {\pi}r^3\\ &= \frac {4\pi}3 (\frac 1{4\pi})^3\\ &= \frac 13 \times \frac 1{16\pi^2}\\ &= \frac 1{44\pi^2}cm^3_{Ans}\\ \end{align*}
  
  Here,
  
  r1 = $$\frac {6cm}2$$ = 3cm
  
  r2 = $$\frac {8cm}2$$ = 4cm
  
  r3 = $$\frac {10cm}{2}$$ = 5cm
  
  Suppose,
  
  radius of new sphere = R
  
  Now,
  
  Volume of new sphere = sum of the volume of first, second and third spheres
  
  or, $$\frac 43$$$${\pi}R^3$$ = $$\frac 43$$$${\pi}r_1^3$$ +$$\frac 43$$$${\pi}r_2^3$$ +$$\frac 43$$$${\pi}r_3^3$$
  
  or, R3 = r13 + r23 + r33
  
  or, R3 = (33 + 43 + 53)
  
  or, R3 = 27 + 64 + 125
  
  or, R3 = 216
  
  or, R = $$\sqrt [3]{216}$$
  
  ∴ R = 6 cm
  
  Hence, the diameter of new sphere (d) = 2R = 2 $$\times$$ 6 cm = 12 cmAns
  
  \begin{align*} \text{Volume of small ball} &= \frac 43 {\pi}r^3\\ &= \frac 43 {\pi}1^3\\ &= \frac 43 {\pi}cm^3\\ \end{align*}
  
  \begin{align*} \text{Radius of big ball} &= \frac {8cm}2\\ &= 4 cm\\ \end{align*}
  
  \begin{align*} \text{Volume of big ball} &= \frac 43 {\pi}r^3\\ &= \frac 43 {\pi}4^3\\ &= \frac 43 {\pi}64cm^3\\ \end{align*}
  
  Now,
  
  \begin{align*} \text{Number of small balls that can be made from the big ball} &= \frac {\frac 43 {\pi} \times 64}{\frac 43 {\pi}}\\ &= 64_{Ans}\\ \end{align*}
  
  Let x be the radius of the first sphere.
  
  Then,
  
  Radius of second sphere is $$\frac r4$$
  
  Volume of first sphere (V1) = $$\frac 43$$ $${\pi}r^3$$
  
  Volume of second sphere (V2) = $$\frac 43$$ $$\pi$$ ($$\frac r4$$)3 = $$\frac {4\pi}{3}$$$$\times$$ $$\frac {r^3}{64}$$
  
  Now,
  
  $$\frac {V_2}{V_1}$$ = $$\frac {\frac {4\pi}{3} \times \frac {r^3}{64}}{\frac 43 {\pi}r^3}$$ = $$\frac 1{64}$$
  
  Hence, V1 :V2 = 1 : 64Ans