Question

Some spherical balls of diameter $2.8\,\text{cm}$ are dropped into a cylindrical container containing some water and are fully submerged. The diameter of the container is $14\,\text{cm}$. Find how many balls have been dropped in it if the water rises by $11.2\,\text{cm}$.

• A. 50
• B. 150
• C. 250
• D. 350

Hint:

Use displacement: rise in cylinder volume $=\pi R^2 h$ equals $n$ times sphere volume $\frac{4}{3}\pi r^3$. The $\pi$ cancels, keeping arithmetic clean.

Answer 1

The Correct Option is B

Step 1: Volumes displaced.
Rise in water gives cylinder volume increase:
Container radius $R=\dfrac{14}{2}=7\,\text{cm}$, rise $h=11.2\,\text{cm}$.
\[ V_{\text{rise}}=\pi R^2 h=\pi\cdot 7^2\cdot 11.2=\pi\cdot 49\cdot 11.2=548.8\,\pi\ \text{cm}^3. \] Step 2: Volume of one sphere.
Ball radius $r=\dfrac{2.8}{2}=1.4\,\text{cm}$.
\[ V_{\text{sphere}}=\frac{4}{3}\pi r^3=\frac{4}{3}\pi(1.4)^3=\frac{4}{3}\pi(2.744)=\frac{1372}{375}\pi\ \text{cm}^3. \] Step 3: Number of balls.
Let $n$ be the number of balls. Then \[ n=\frac{V_{\text{rise}}}{V_{\text{sphere}}} =\frac{548.8\,\pi}{\tfrac{1372}{375}\pi} =\frac{548.8\times 375}{1372} =\frac{\tfrac{2744}{5}\times 375}{1372} =\frac{2744}{1372}\times \frac{375}{5} =2\times 75=150. \] \[ \boxed{150} \]