A sealed bottle, which contains water, has been constructed by attaching a cylinder of radius $1 \mathrm{~cm}$ to a cylinder of radius $3 \mathrm{~cm}$, as shown in Figure A. When the bottle is right side up, the height of the water inside is $20 \mathrm{~cm}$, as shown in the cross-section of the bottle in Figure B. When the bottle is upside down, the height of the liquid is $28 \mathrm{~cm}$, as shown in Figure C. What is the total height, in cm, of the bottle?

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-135.jpg?height=301&width=347&top_left_y=2153&top_left_x=499)

Figure A

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-135.jpg?height=282&width=353&top_left_y=2185&top_left_x=905)

Figure B

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-135.jpg?height=279&width=352&top_left_y=2186&top_left_x=1301)

Figure C
(A) 29
(B) 30
(C) 31
(D) 32
(E) 48

continued ...
A sealed bottle, which contains water, has been constructed by attaching a cylinder of radius $1 \mathrm{~cm}$ to a cylinder of radius $3 \mathrm{~cm}$, as shown in Figure A. When the bottle is right side up, the height of the water inside is $20 \mathrm{~cm}$, as shown in the cross-section of the bottle in Figure B. When the bottle is upside down, the height of the liquid is $28 \mathrm{~cm}$, as shown in Figure C. What is the total height, in $\mathrm{cm}$, of the bottle?

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-268.jpg?height=342&width=371&top_left_y=585&top_left_x=430)

Figure A

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-268.jpg?height=304&width=391&top_left_y=623&top_left_x=867)

Figure B

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-268.jpg?height=301&width=390&top_left_y=625&top_left_x=1320)

Figure C
(A) 29
(B) 30
(C) 31
(D) 32
(E) 48

## Solution

We'll start by representing the height of the large cylinder as $h_{1}$ and the height of the small cylinder as $h_{2}$. For simplicity, we'll let $x=h_{1}+h_{2}$.

If the bottom cylinder is completely filled and the top cylinder is only partially filled the top cylinder will have a cylindrical space that is not filled. This cylindrical space will have a height equal to $x-20$ and a volume equal to, $\pi(1)^{2}(x-20)$.

Similarly, if we turn the cylinder upside down there will be a cylindrical space unfilled that will have a height equal to $x-28$ and a volume equal to, $\pi(3)^{2}(x-28)$.

Since these two unoccupied spaces must be equal, we then have,

$$
\begin{aligned}
\pi(1)^{2}(x-20) & =\pi(3)^{2}(x-28) \\
x-20 & =9 x-252 \\
8 x & =272 \\
x & =29
\end{aligned}
$$

Therefore, the total height is 29 .

ANSWER: (A)