Chapter 7.3, Problem 87PPS

To determine

To calculate the possible volumes of each size soccer ball.

Answer to Problem 87PPS

size 3: 204.0 to 230.2 cubic inches

size 4: 268.5 to 299.9 cubic inches

size 5: 333.6 to 382.4 cubic inches

Explanation of Solution

Given:

The radius r of a ball that holds V cubic units of air is modeled by $r=0.62V^{\frac{1}{3}}.$

  

Calculation:

Given that, the radius r of a ball that holds V cubic units of air is modeled by $r=0.62V^{\frac{1}{3}}.$

Substituting each size and radius of the ball the volumes can be calculated.

Size 3:

  $d=7.3\Rightarrow r=3.65$

  $\begin{array}{c}r=0.62V^{\frac{1}{3}}\\ 3.65=0.62V^{\frac{1}{3}}\end{array}$

Dividing 0.62 on both sides and simplifying,

  $\begin{array}{l}\frac{3.65}{0.62}=\frac{\cancel{0.62}}{\cancel{0.62}}{V}^{\frac{1}{3}}\\ \frac{3.65}{0.62}=V^{\frac{1}{3}}\end{array}$

Cubing on both sides and simplifying,

  $\begin{array}{c}{\left(\frac{3.65}{0.62}\right)}^3={\left(V^{\frac{1}{3}}\right)}^3\\ {\left(\frac{3.65}{0.62}\right)}^3=V\\ 204.0\simeq V\end{array}$

Hence, the volume of the soccer ball of size 3 is approximately 204.0.

Size 3:

  $d=7.6\Rightarrow r=3.8$

  $\begin{array}{c}r=0.62V^{\frac{1}{3}}\\ 3.8=0.62V^{\frac{1}{3}}\end{array}$

Dividing 0.62 on both sides and simplifying,

  $\begin{array}{l}\frac{3.8}{0.62}=\frac{\cancel{0.62}}{\cancel{0.62}}{V}^{\frac{1}{3}}\\ \frac{3.8}{0.62}=V^{\frac{1}{3}}\end{array}$

Cubing on both sides and simplifying,

  $\begin{array}{c}{\left(\frac{3.8}{0.62}\right)}^3={\left(V^{\frac{1}{3}}\right)}^3\\ {\left(\frac{3.8}{0.62}\right)}^3=V\\ 230.2\simeq V\end{array}$

Hence, the volume of the soccer ball of size 3 is approximately 230.2.

Size 4:

  $d=8.0\Rightarrow r=4.0$

  $\begin{array}{c}r=0.62V^{\frac{1}{3}}\\ 4.0=0.62V^{\frac{1}{3}}\end{array}$

Dividing 0.62 on both sides and simplifying,

  $\begin{array}{l}\frac{4.0}{0.62}=\frac{\cancel{0.62}}{\cancel{0.62}}{V}^{\frac{1}{3}}\\ \frac{4.0}{0.62}=V^{\frac{1}{3}}\end{array}$

Cubing on both sides and simplifying,

  $\begin{array}{c}{\left(\frac{4}{0.62}\right)}^3={\left(V^{\frac{1}{3}}\right)}^3\\ {\left(\frac{4}{0.62}\right)}^3=V\\ 268.5\simeq V\end{array}$

Hence, the volume of the soccer ball of size 3 is approximately 268.5.

Size 4:

  $d=8.3\Rightarrow r=4.15$

  $\begin{array}{c}r=0.62V^{\frac{1}{3}}\\ 4.15=0.62V^{\frac{1}{3}}\end{array}$

Dividing 0.62 on both sides and simplifying,

  $\begin{array}{l}\frac{4.15}{0.62}=\frac{\cancel{0.62}}{\cancel{0.62}}{V}^{\frac{1}{3}}\\ \frac{4.15}{0.62}=V^{\frac{1}{3}}\end{array}$

Cubing on both sides and simplifying,

  $\begin{array}{c}{\left(\frac{4.15}{0.62}\right)}^3={\left(V^{\frac{1}{3}}\right)}^3\\ {\left(\frac{4.15}{0.62}\right)}^3=V\\ 299.9\simeq V\end{array}$

Hence, the volume of the soccer ball of size 3 is approximately 299.9.

Size 5:

  $d=8.6\Rightarrow r=4.3$

  $\begin{array}{c}r=0.62V^{\frac{1}{3}}\\ 4.3=0.62V^{\frac{1}{3}}\end{array}$

Dividing 0.62 on both sides and simplifying,

  $\begin{array}{l}\frac{4.3}{0.62}=\frac{\cancel{0.62}}{\cancel{0.62}}{V}^{\frac{1}{3}}\\ \frac{4.3}{0.62}=V^{\frac{1}{3}}\end{array}$

Cubing on both sides and simplifying,

  $\begin{array}{c}{\left(\frac{4.3}{0.62}\right)}^3={\left(V^{\frac{1}{3}}\right)}^3\\ {\left(\frac{4.3}{0.62}\right)}^3=V\\ 333.6\simeq V\end{array}$

Hence, the volume of the soccer ball of size 3 is approximately 333.6.

Size 5:

  $d=9.0\Rightarrow r=4.5$

  $\begin{array}{c}r=0.62V^{\frac{1}{3}}\\ 4.5=0.62V^{\frac{1}{3}}\end{array}$

Dividing 0.62 on both sides and simplifying,

  $\begin{array}{l}\frac{4.5}{0.62}=\frac{\cancel{0.62}}{\cancel{0.62}}{V}^{\frac{1}{3}}\\ \frac{4.5}{0.62}=V^{\frac{1}{3}}\end{array}$

Cubing on both sides and simplifying,

  $\begin{array}{c}{\left(\frac{4.5}{0.62}\right)}^3={\left(V^{\frac{1}{3}}\right)}^3\\ {\left(\frac{4.5}{0.62}\right)}^3=V\\ 382.4\simeq V\end{array}$

Hence, the volume of the soccer ball of size 3 is approximately 382.4.

Conclusion:

Thus, the volume of size 3, size 4 and size 5 varies from 204.0 to 230.2 cubic inches, 268.5 to 299.9 cubic inches and 333.6 to 382.4 cubic inches.