Maths M.C.Q

Given that,
Curved surface area $= 264 \mathrm{~m}^2$
Volume $= 924 \mathrm{~m}^3$
Now,
Curved surface area $=2\pi\text{rh}$
$\Rightarrow264=2\times\frac{22}{7}\times\text{r}\times\text{h}$
$\Rightarrow264=\frac{44}{7}\times\text{r}\times\text{h}$
$\Rightarrow\text{r}=\frac{264\times7}{44\times\text{h}}$
$\Rightarrow\text{r}=\frac{42}{\text{h}}$
Volume $=\pi\text{r}^2\text{h}$
$\Rightarrow924=\frac{22}{7}\times\frac{42}{\text{h}}\times\frac{42}{\text{h}}\times\text{h}$
$\Rightarrow924=22\times\frac{6}{\text{h}}\times42$
$\Rightarrow\frac{924}{22\times42\times6}=\frac{1}{\text{h}}$
$\Rightarrow\frac{924}{5544}=\frac{1}{\text{h}}$
$\Rightarrow\text{h}=\frac{5544}{924}$
$\Rightarrow\text{h}=6\text{m}$

Lateral surface area of the cuboid $= 2(l + b) × h$
$=2(15+6)\times\frac{5}{10}$ $...\Big(1\text{dm}=\frac{1}{10}\text{m}\Rightarrow5\text{dm}=\frac{5}{10}\text{m}\Big)$
$=2(21)\times\frac{1}{2}$
$=21\text{m}^2$

Let their heights be $x\ cm$ and $2x\ cm$ respectively and let their radii be $\text{R}_1$ and $\text{R}_2$ respectively.
$\Rightarrow\pi\text{R}_1^2\text{h}=\pi\text{R}_2^2\text{h}$
$\Rightarrow\pi\times\text{R}_1^2\times\text{x}=\pi\times\text{R}_2^2\times2\text{x}$
$\Rightarrow\text{R}_1^2=\text{R}_2^2\times2$
$\Rightarrow\Big(\frac{\text{R}_1}{\text{R}_2}\Big)^2=2$
$\Rightarrow\frac{\text{R}_1}{\text{R}_2}=\frac{\sqrt2}{1}$
$⇒$ Ratio of their radii is $\sqrt2:1$

Let the side of the cube be $x$.
Volume of new cube formed $=3^3+4^3+5^3$
$\Rightarrow x^3=27+64+125$
$\Rightarrow \mathrm{x}^3=216 \mathrm{~cm}^3$
$\Rightarrow x=6 \mathrm{~cm}$
Lateral surface area of the new cube $=4 \mathrm{x}^2=4 \times(6)^2=144 \mathrm{~cm}^2$

Let $r_1$ and $r_2$ be the radii and $h_1$ and $h_2$ be the height of two cylinders.
$\Rightarrow\frac{\text{r}_1}{\text{r}_2}=\frac{2}{3}\ \text{and}\ \frac{\text{h}_1}{\text{h}_2}=\frac{5}{3}$
Ratio of the surface area $=\frac{2\pi\text{r}_1\text{h}_1}{2\pi\text{r}_2\text{h}_2}$
$=\frac{\text{r}_1}{\text{r}_2}\times\frac{\text{h}_1}{\text{h}_2} $
$=\frac{2}{3}\times\frac{5}{3}$
$=\frac{10}{9}$
$=10:9$