Chapter 1, Problem 41GP

To determine

The number of jelly beans in the jar.

Answer to Problem 41GP

Solution:

The number of candies in the jar is 130.

Explanation of Solution

Given:

A cylindrical jar filled with cylindrical shape candies

Let the radius of the jar is $R=5\text{cm}$

The height of the jar is $H=15\text{cm}$

The radius of the candy is $r=1\text{cm}$

The height of the candy, $h=2.5\text{cm}$

Formula used:

The volume of a cylinder is given as

  $V=\pi R^{2}H$

Here,

  $R$ is radius

  $H$ is the height

Number of candies in the jar is calculated as

  $n=\frac{\text{Volume}\text{of}\text{jar}}{\text{Volume}\text{of}\text{one}\text{candy}}$

Calculation:

The volume of the jar is calculated as

  $\begin{array}{c}{V}_{\text{jar}}=\pi \times {0.05}^2\times 0.15\\ =0.0011{\text{m}}^{\text{3}}\end{array}$

The candies are of irregular shape and due to the irregularity of the candies,a fraction of the volume of the jar is occupied by the air, let the volume of available for the candies is 90% of the total volume of the jar.

The effective volume of the jar is calculated as

  $\begin{array}{c}{V}_{\text{effective}}=0.9\times 0.0011\\ =0.00099{\text{m}}^{\text{3}}\end{array}$

The volume of one candy is calculated as

  $\begin{array}{c}{V}_{\text{candy}}=\pi \times {0.01}^2\times 0.025\\ =7.85\times {10}^{-6}{\text{m}}^{\text{3}}\end{array}$

Number of candies in the jar is calculated as

  $\begin{array}{c}n=\frac{0.00099}{7.85\times {10}^{-6}}\\ =126.11\\ =130\end{array}$

Conclusion:

The number of candies in a jaris 130.