Chapter 12.6, Problem 9PSB

To determine

To calculate: The volume of medicine in a capsule.

Answer to Problem 9PSB

The volume of medicine in a given capsule is $91.891{\text{ mm}}^{\text{3}}.$

Explanation of Solution

Given information: A cold capsule is of length $\left(l\right)=11\text{ mm}$ and diameter $\left(d\right)=3\text{ mm}$ .

Formula used:

Volume of Cylinder $\left(V\right)=\pi r^{2}l$ ……………………… (i)

where r = radius of the base or top of cylinder

l = length of the cylinder

Volume of Hemisphere $\left(V\right)=\left(\frac{2}{3}\right)\pi r^3$ ………………. (ii)

where r = radius of the hemisphere

Calculation: As we can see that the capsule is just a cylinder enclosed by two identical hemispheres at both the ends. So,

Total Volume of medicine in a capsule $\left(V\right)=$ Volume of Cylinder $+2\times$ Volume of Hemisphere

Now using equation (i) and (ii) in above equation to get,

  $V=\pi r^{2}l+2\times \left(\frac{2}{3}\right)\pi r^3$ ………………………..……… (iii)

Both the solids must have equal radius as common base is being shared in between.

Substituting the values $r=\frac{d}{2}=\frac{3}{2}=1.5\text{ mm}$ and $l=11\text{ mm}$ in equation (iii)

  $\begin{array}{l}V=\left(\pi \times {1.5}^2\times 11\right){\text{ mm}}^{\text{3}}+\left\{2\times \left(\frac{2}{3}\right)\times \pi \times {1.5}^3\right\}{\text{ mm}}^{\text{3}}\\ V=77.754{\text{ mm}}^{\text{3}}+14.137{\text{ mm}}^{\text{3}}\\ V=91.891{\text{ mm}}^{\text{3}}\end{array}$

Hence, the volume of medicine in a capsule is $91.891{\text{ mm}}^{\text{3}}.$