Volume And Surface Area

Details

External radius = 6 cm
Internal radius = (6 - 0.25) = 5.75 cm
Volume of material in hollow cylinder :
$$ = \left[ {\frac{{22}}{7} \times \left\{ {{{\left( 6 \right)}^2} - {{\left( {5.75} \right)}^2}} \right\} \times 15} \right]{\text{c}}{{\text{m}}^3}$$
$$ = \left( {\frac{{22}}{7} \times 11.75 \times 0.25 \times 15} \right){\text{c}}{{\text{m}}^3}$$
$$ = \left( {\frac{{22}}{7} \times \frac{{1175}}{{100}} \times \frac{{25}}{{100}} \times 15} \right){\text{c}}{{\text{m}}^3}$$
$$ = \left( {\frac{{11 \times 705}}{{56}}} \right){\text{c}}{{\text{m}}^3}$$
Let the length of solid cylinder be h
Then,
$$\frac{{22}}{7} \times 1 \times 1 \times h = \left( {\frac{{11 \times 705}}{{56}}} \right)$$
$$ ⇒ h = \left( {\frac{{11 \times 705}}{{56}} \times \frac{7}{{22}}} \right)$$
$$ ⇒ h = 44.0625\,{\text{cm}}$$

Let the original radius and height of the cone be r and h respectively
Then, Original volume = $$\frac{1}{3}\pi {r^2}h$$
New radius = $$\frac{r}{2}$$ and new hight = 2h
New volume :
$$ = \frac{1}{3} \times \pi \times {\left( {\frac{r}{2}} \right)^2} \times 3h$$
$$ = \frac{3}{4} \times \frac{1}{3}\pi {r^2}h$$
∴ Decrease % :
$$ = \left( {\frac{{\frac{1}{4} \times \frac{1}{3}\pi {r^2}h}}{{\frac{1}{3}\pi {r^2}h}} \times 100} \right)\% $$
$$ = 25\% $$

Let the length of each side of the cube be a cm
Then, Volume of the part of cube outside water = Volume of the mass placed on it
⇒ 2a² = 0.2 × 1000
⇒ 2a² = 200
⇒ a² = 100
⇒ a = 10

Details

Details

Details

Let the length of the wire be h
Then,
$$\pi \times \frac{3}{{20}} \times \frac{3}{{20}} \times h = \frac{4}{3}\pi \times 4 \times 4 \times 4$$
$$ ⇒ h = \left( {\frac{{4 \times 4 \times 4 \times 4 \times 20 \times 20}}{{3 \times 3 \times 3}}} \right)cm$$
$$ ⇒ h = \left( {\frac{{102400}}{{27}}} \right)cm$$
$$ ⇒ h = 3792.5\,cm$$
$$ ⇒ h = 37.9\,m$$

Internal radius, r = 4 cm
External radius, R = 5 cm
Total surface area :
$$ = 2\pi {R^2} + 2\pi {r^2} + \pi \left( {{R^2} - {r^2}} \right)$$
$$ = 3\pi {R^2} + \pi {r^2}$$
$$ = \left[ {\pi \left( {3 \times 25 + 16} \right)} \right]{\text{ c}}{{\text{m}}^2}$$
$$ = \left( {\frac{{22}}{7} \times 91} \right){\text{c}}{{\text{m}}^2}$$
$$ = 286{\text{ c}}{{\text{m}}^2}$$

$$l = 10\,m,h = 8\,m$$
$$So,$$
$$r = \sqrt {{l^2} + {h^2}} $$
$$\,\,\,\, = \sqrt {{{\left( {10} \right)}^2} - {{\left( 8 \right)}^2}} $$
$$\,\,\,\, = 6\,m$$
∴ Curved surface area :
$$ = \pi rl$$
$$ = \left( {\pi \times 6 \times 10} \right){m^2}$$
$$ = 60\pi \,{m^2}$$