Question

Estimate the number of jelly beans in the jar of Fig. 1–18.

FIGURE 1-18 Problem 45. Estimate the number of jelly beans in the jar.

Short answer

There are 400 jelly beans in the jar.

Step-by-step solution

Step 1. Formula for the number of jelly beans

The expression for estimating the number of jelly beans in the jar is:

$N=\frac{V_j}{V_b}$

Here,$V_j$is the volume of the jar and$V_b$is the volume of the jelly bean.

Step 2. Calculation of the volume of the jelly bean

Let the length of the cylindrical jelly bean be approximately $h=2.0\mathrm{cm}$, and the radius be $r=0.63\mathrm{cm}$. Then, the volume of the jelly bean jar will be:

$\begin{array}{l}{V}_b=\pi r^{2}h\\ V_b=\pi {\left(0.63\mathrm{cm}\right)}^3\left(2.0\mathrm{cm}\right)\\ V_b=2.49{\mathrm{cm}}^3\end{array}$

Suppose the volume of the jar is $V_j=1110{\mathrm{cm}}^3$. Due to the irregular shape of the jelly beans, they are not tightly packed. So, approximately 90% volume of the jar is filled.

The effective volume of the jar is:

$\begin{array}{l}{V}_{e,j}=90\%\left(V\right)\\ V_{e,j}=\left(\frac{90}{100}\right)\left(1110{\mathrm{cm}}^3\right)\\ V_{e,j}=999{\mathrm{cm}}^3\end{array}$

Step 3. Calculation of the number of jelly beans in the jar

The number of jelly beans in the jar can be calculated as:

$\begin{array}{l}N=\frac{V_{e,j}}{V_b}\\ N=\frac{999{\mathrm{cm}}^3}{2.49{\mathrm{cm}}^3}\\ N\approx 400\end{array}$

Thus, there are 400 jelly beans in the jar.