Question

What is the difference in volume between a baseball with a diameter of $1.75in$ and a soccer ball with a diameter of $9.50in$?

Answer 1

Hint: We solve this problem by using the simple formula of volume of a sphere. The volume of a sphere of radius $'r'$ is given as,
$V=\dfrac{4}{3}\pi {{r}^{3}}$
By using this formula we find the volume of both baseball and soccer ball and then we subtract the smaller value from the bigger value to get the required answer.

Complete step by step solution:
We are asked to find the difference in volume of given baseball and soccer ball.
Let us find the volume of baseball first.
We are given that the diameter of the baseball is $1.75in$
We know that the radius is half of the diameter. By using this condition we get the radius of the baseball as,
$\begin{align}
  & \Rightarrow {{r}_{b}}=\dfrac{1.75}{2} \\
 & \Rightarrow {{r}_{b}}=0.875in \\
\end{align}$
Now, let us find the volume of the baseball.
We know that the baseball is in the shape of a sphere and the formula of volume of sphere of radius $'r'$ is given as,
$V=\dfrac{4}{3}\pi {{r}^{3}}$
By using this formula of volume we get the volume of baseball as,
$\Rightarrow {{V}_{b}}=\dfrac{4}{3}\pi {{\left( 0.875 \right)}^{3}}$
We know that the value of $\pi $ up to 2 decimal points is $3.14$
By using the value of $\pi $ in the volume then we get,
$\begin{align}
  & \Rightarrow {{V}_{b}}=\dfrac{4\times 3.14\times {{\left( 0.875 \right)}^{3}}}{3} \\
 & \Rightarrow {{V}_{b}}=\dfrac{8.414}{3} \\
 & \Rightarrow {{V}_{b}}=2.80cu.in \\
\end{align}$
Now, let us find the volume of soccer ball.
We are given that the diameter of the soccer ball is $9.5in$
We know that the radius is the half of diameter. By using this condition we get the radius of the soccer ball as,
$\begin{align}
  & \Rightarrow {{r}_{s}}=\dfrac{9.5}{2} \\
 & \Rightarrow {{r}_{s}}=4.75in \\
\end{align}$
Now, let us find the volume of the baseball.
By using this formula of volume we get the volume of soccer ball as,
$\Rightarrow {{V}_{s}}=\dfrac{4}{3}\pi {{\left( 4.75 \right)}^{3}}$
We know that the value of $\pi $ up to 2 decimal points is $3.14$
By using the value of $\pi $ in the volume then we get,
$\begin{align}
  & \Rightarrow {{V}_{s}}=\dfrac{4\times 3.14\times {{\left( 4.75 \right)}^{3}}}{3} \\
 & \Rightarrow {{V}_{s}}=\dfrac{1346.078}{3} \\
 & \Rightarrow {{V}_{s}}=448.69\simeq 448.70cu.in \\
\end{align}$
Now, let us find the difference between the two volumes then we get,
$\begin{align}
  & \Rightarrow V={{V}_{s}}-{{V}_{b}} \\
 & \Rightarrow V=448.70-2.80 \\
 & \Rightarrow V=445.9\simeq 446cu.in \\
\end{align}$
Therefore we can conclude that the difference of the volume of the given baseball and the soccer ball is given as 446 cubic inches that is,
$\therefore V=446cu.in$

Note: We need to be very careful in doing the calculations because there seem to be more calculations while solving the volume of the balls.
Also we need to take care that the diameter of the balls is given but not the radius. Sometimes students may assume that the given value is radius and calculate the volume that leads to wrong answer. Also we need to take care of units. The diameter is given in inches so that the volume will be cubic inches. If there is any need for conversion from inches to centi meters then the conversion is given as,
$1in=2.56cm$