Question

A sealed bottle containing some water is made up of two cylinders $A$ and $B$ of radius $1.5cm$ and $3cm$ respectively, as shown in the figure. When the bottle is placed right up on a table, the height of water in it is $15cm$, but when placed upside down, the height of water is $24cm$. The height of the bottle is

• A. $25cm$
• B. $26cm$
• C. $27cm$
• D. $28cm$

Answer 1

Answer: (C)

$27cm$

Let heights of cylinder $A$ and $B$ are $h_1$ and $h_2$ respectively.
Volume in first situation $=\pi \times (3{)}^2\times h_2+\pi \times (1.5{)}^2\times (15-h_2)$

$=\pi [h_2\left\{\right(3{)}^2-\left(1.5{)}^2\right\}+\{15\times (1.5{)}^2\}]$
Volume in second situation $=\pi \times (1.5{)}^2\times h_1+\pi \times (3{)}^2\times (24-h_1)$

$=\pi [h_1\left\{\right(1.5{)}^2-\left(3{)}^2\right\}+\{24\times (3{)}^2\}]$
Volume is same
$\Rightarrow [h_2\left\{\right(3{)}^2-\left(1.5{)}^2\right\}+\{15\times (1.5{)}^2\}]=\pi [h_1\left\{\right(1.5{)}^2-\left(3{)}^2\right\}+\{24\times (3{)}^2\}]$

$\Rightarrow (h_1+h_2)\left\{\right(3{)}^2-\left(1.5{)}^2\right\}24\times (3{)}^2-15\times (1.5{)}^2$

$\Rightarrow$ Total height $=h_1+h_2=\frac{3\times \left(1.5{)}^2\right(32-5)}{3\times (1.5{)}^2}=27$.