Q. 63 - Calculus

Question

Alan makes beads that have the shape of flattened spheres with holes drilled through for stringing. The beads can be modeled by revolving the curve $f\left(x\right)=$ $3-0.15x^2$ on $[-4,4]$ around the x-axis, as shown in the next figure, in which units are in centimeters. Use the shell method to determine the amount of material required to make 100 beads of this shape.

Step-by-Step Solution

Answer

The material needed for 100 beads is $12546.3{ \mathrm{cm}}^3$.

Step 1: Given information

Consider the given figure and the bead's model equation is, $f\left(x\right)=3-0.15 x^2$, $x\in [-4,4]$.

Step 2: Calculate the required value.

The goal is to find the material needed for 100 beads. 

The bead's outer radius is 3 cm and the bead's inner radius can be calculated as,

$\begin{array}{rcl}y& =& 3-0.15(4{)}^2\\ & =& 3-2.4\\ & =& 0.6 \mathrm{cm}\end{array}$

A bead's volume is computed as follows:

style="width:30%" $\begin{array}{rcl}V& =& 2\times 2\pi {\int }_{0.6}^{3}y\sqrt{\frac{3-y}{0.15}}dy\\ & =& \frac{4\pi }{\sqrt{0.15}}{\int }_{0.6}^{3}y\sqrt{3-y}dy\end{array}$

Simplify the above integral as follows:

style="width:30%" $\begin{array}{rcl}V& =& \frac{4\pi }{\sqrt{0.15}}{\left[\frac{2}{5}(3-x{)}^{5/2}-2(3-x{)}^{3/2}\right]}_{0.6}^3\\ & =& \frac{4\pi }{\sqrt{0.15}}\left[-\frac{2}{5}(3-0.6{)}^{5/2}+2(3-0.6{)}^{3/2}\right]\\ & \approx & 125.463{ \mathrm{cm}}^3\end{array}$

1 bead has a volume of roughly $125.463$ cubic centimeters.

Hence, the volume of 100 beads isstyle="width:30%" $125.463\times 100=12546.3$.

As a result, the material needed for 100 beads isstyle="width:30%" $12546.3{ \mathrm{cm}}^3$.