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Sagot :
Answer:
8.96 in³
Step-by-step explanation:
In order to solve this, we need to find the volume of the entire can. From there we can find the volume of each tennis ball and multiply that value by 3 in order to account for all the tennis balls. From there we subtract the tennis balls' volume by the can's volume to get the volume of empty space.
Volume formula for tennis can:
[tex]\begin{document}\fbox{\begin{minipage}{\textwidth} The volume \( V \) of a cylinder is given by: \[ V = \pi r^2 h \] Where: \begin{itemize} \item \( r \) is the radius of the base of the cylinder, \item \( h \) is the height of the cylinder. \end{itemize}\end{minipage}}\end{document}[/tex]
Volume formula for tennis ball:
[tex]\begin{document}\fbox{\begin{minipage}{\textwidth} The volume \( V \) of a sphere is given by: \[ V = \frac{4}{3} \pi r^3 \] Where: \begin{itemize} \item \( r \) is the radius of the sphere. \end{itemize}\end{minipage}}\end{document}[/tex]
Solving:
Volume of Can:
[tex]\text{Radius of the can, \( r_{\text{can}} \)}: \[ r_{\text{can}} = \frac{\text{diameter}}{2} = \frac{2.25}{2} = \boxed{1.125 \, \text{inches}} \]\\ \item Height of the can, \( h_{\text{can}} \): \[ \boxed{ h_{\text{can}} = 6.75 \, \text{inches}} \] \item Volume of the can, \( V_{\text{can}} \): \[ V_{\text{can}} = \pi r_{\text{can}}^2 h_{\text{can}} \][/tex]
[tex]V_{\text{can}} = 3.14 \times (1.125)^2 \times 6.75\\ V_{\text{can}} = 3.14 \times 1.266 \times 6.75\\ V_{\text{can}} \approx 3.14 \times 8.555\\ \boxed{V_{\text{can}} \approx 26.84 \, \text{cubic inches}}[/tex]
[tex]\hrulefill[/tex]
Volume of Ball:
[tex]\[ r_{\text{ball}} = \frac{\text{diameter}}{2} = \frac{2.25}{2} =\boxed{ 1.125 \, \text{inches}}\\ \] \item Volume of one ball, \( V_{\text{ball}} \): \[ V_{\text{ball}} = \frac{4}{3} \pi r_{\text{ball}}^3 \][/tex]
[tex]V_{\text{ball}} = \frac{4}{3} \times 3.14 \times (1.125)^3\\ V_{\text{ball}} = \frac{4}{3} \times 3.14 \times 1.422\\ V_{\text{ball}} \approx \frac{4}{3} \times 4.47\\ \boxed{V_{\text{ball}} \approx 5.96 \, \text{cubic inches}}[/tex]
Since there are 3 balls, find the total volume by multiplying the value by 3:
[tex]V_{\text{total balls}} = 3 \times V_{\text{ball}}\\ V_{\text{total balls}} = 3 \times 5.96\\ \boxed{ V_{\text{total balls}} = 17.88 \, \text{cubic inches}}[/tex]
[tex]\hrulefill[/tex]
Volume of Empty Space:
[tex]V_{\text{empty}} = V_{\text{can}} - V_{\text{total balls}}\\ V_{\text{empty}} = 26.84 - 17.88\\ \boxed{V_{\text{empty}} = 8.96 \, \text{cubic inches}}[/tex]
[tex]\hrulefill[/tex]
Therefore, the area of empty space is 8.96 cubic inches( in³).
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