Question

A spherical glass vessel has a cylindrical neck $8$ cm long, $2$ cm in diameter, and the diameter of the spherical part is $8.5$ cm. Find the volume of water that can be filled in it.

Hint:

Model the vessel as "sphere $+$ cylinder." Use $V_{\text{sphere}}=\tfrac{4}{3}\pi R^{3}$ and $V_{\text{cyl}}=\pi r^{2}h$, then add.

Answer 1

For the neck (cylinder): radius $r_1=1$ cm, height $h=8$ cm; $\ V_1=\pi r_1^{2}h=\pi\cdot1^{2}\cdot8=8\pi\ \text{cm}^{3}$.
For the bulb (sphere): radius $R=\dfrac{8.5}{2}=4.25$ cm; $\ V_2=\dfrac{4}{3}\pi R^{3} =\dfrac{4}{3}\pi(4.25)^{3}=\dfrac{4913}{48}\pi\ \text{cm}^{3}$.
Total volume to fill: \[ V=V_1+V_2=\left(8+\frac{4913}{48}\right)\pi=\frac{5297}{48}\pi\ \text{cm}^{3}. \] Taking $\pi=\dfrac{22}{7}$, \[ V=\frac{5297}{48}\cdot\frac{22}{7}=\frac{116534}{336}\approx 346.82\ \text{cm}^{3}. \] \[ \boxed{\text{Volume}\approx 346.8\ \text{cm}^{3}} \]