Surface Areas and Volumes — Important Questions

Answer: (a) $462\ \text{cm}^2$

TSA of a solid hemisphere $=3\pi r^2=3\times\dfrac{22}{7}\times 49=462\ \text{cm}^2.$

Answer: (a) $27$

Number $=\dfrac{\frac{4}{3}\pi(3)^3}{\frac{4}{3}\pi(1)^3}=\dfrac{27}{1}=27.$

Answer: (a) $550\ \text{cm}^2$

$l=\sqrt{7^2+24^2}=\sqrt{625}=25\ \text{cm}.$ CSA $=\pi rl=\dfrac{22}{7}\times 7\times 25=550\ \text{cm}^2.$

Answer: (a) $18\ \text{cm}$

$\dfrac{4}{3}\pi(6)^3=\pi(4)^2h\Rightarrow 288\pi=16\pi h\Rightarrow h=18\ \text{cm}.$

Answer: (d) A is false but R is true.

The curved surface area of a hemisphere is $2\pi r^2$ (R is true), but its total surface area also includes the flat circular base, giving $2\pi r^2+\pi r^2=3\pi r^2$; hence A is false.

$\dfrac{\text{Volume of cone}}{\text{Volume of cylinder}}=\dfrac{\frac{1}{3}\pi r^2h}{\pi r^2h}=\dfrac{1}{3}$, so the ratio is $1:3.$

Volume of cone $=\dfrac{1}{3}\times 360=120\ \text{cm}^3.$

Each cube has side $=\sqrt[3]{64}=4\ \text{cm}.$ The cuboid formed has dimensions $8\times 4\times 4\ \text{cm}.$

Surface area $=2(lb+bh+hl)=2(8\times 4+4\times 4+4\times 8)=2(32+16+32)=2\times 80=160\ \text{cm}^2.$

Radius $r=3.5\ \text{cm}.$ Height of cone $=15.5-3.5=12\ \text{cm}.$

Slant height $l=\sqrt{r^2+h^2}=\sqrt{3.5^2+12^2}=\sqrt{12.25+144}=\sqrt{156.25}=12.5\ \text{cm}.$

TSA $=\text{CSA of cone}+\text{CSA of hemisphere}=\pi rl+2\pi r^2=\pi r(l+2r)$
$=\dfrac{22}{7}\times 3.5\times(12.5+7)=11\times 19.5=214.5\ \text{cm}^2.$

Volume of cylinder $=\pi r^2h=\pi\times 6^2\times 15=540\pi\ \text{cm}^3.$

Volume of one cone $+$ hemisphere $=\dfrac{1}{3}\pi(3)^2(12)+\dfrac{2}{3}\pi(3)^3=36\pi+18\pi=54\pi\ \text{cm}^3.$

Number of cones $=\dfrac{540\pi}{54\pi}=10.$

Volume of block $=4.4\times 2.6\times 1=11.44\ \text{m}^3.$

Internal radius $=0.30\ \text{m}$, external radius $=0.30+0.05=0.35\ \text{m}.$

Volume of pipe $=\pi(R^2-r^2)H=\dfrac{22}{7}\big(0.35^2-0.30^2\big)H=\dfrac{22}{7}(0.0325)H.$

Set equal to $11.44$: $\dfrac{22}{7}\times 0.0325\times H=11.44\Rightarrow 0.10214\,H=11.44\Rightarrow H=112\ \text{m}.$

Radius $r=0.7\ \text{cm}$, height $h=2.4\ \text{cm}$, slant height $l=\sqrt{0.7^2+2.4^2}=\sqrt{0.49+5.76}=\sqrt{6.25}=2.5\ \text{cm}.$

(i) TSA $=\text{CSA of cylinder}+\text{area of base}+\text{CSA of cone}=2\pi rh+\pi r^2+\pi rl=\pi r(2h+r+l)$
$=\dfrac{22}{7}\times 0.7\times(4.8+0.7+2.5)=2.2\times 8.0=17.6\ \text{cm}^2.$

(ii) Volume $=\pi r^2h-\dfrac{1}{3}\pi r^2h=\dfrac{2}{3}\pi r^2h=\dfrac{2}{3}\times\dfrac{22}{7}\times 0.49\times 2.4=2.464\ \text{cm}^3.$

Radius of tank $=5\ \text{m}$, depth $=2\ \text{m}.$ Volume of tank $=\pi(5)^2(2)=50\pi\ \text{m}^3.$

Pipe radius $=10\ \text{cm}=0.1\ \text{m}.$ In $1$ hour water flows a length of $3\ \text{km}=3000\ \text{m}.$

Volume of water per hour $=\pi(0.1)^2\times 3000=30\pi\ \text{m}^3.$

Time $=\dfrac{50\pi}{30\pi}=\dfrac{5}{3}\ \text{hours}=\dfrac{5}{3}\times 60=100\ \text{minutes}.$

(i) $l=\sqrt{r^2+h^2}=\sqrt{1.5^2+2^2}=\sqrt{2.25+4}=\sqrt{6.25}=2.5\ \text{m}.$

(ii) Volume $=\pi r^2h_{\text{cyl}}+\dfrac{1}{3}\pi r^2h_{\text{cone}}=\dfrac{22}{7}(1.5)^2(7)+\dfrac{1}{3}\cdot\dfrac{22}{7}(1.5)^2(2)$
$=49.5+4.71=54.21\ \text{m}^3.$

(iii) Curved area $=2\pi rh_{\text{cyl}}+\pi rl=2\times\dfrac{22}{7}\times 1.5\times 7+\dfrac{22}{7}\times 1.5\times 2.5=66+11.79=77.79\ \text{m}^2.$