Respuesta :
Solution:
Given:
The dimensions of the rectangular box are;
[tex]8.8in\times8.5in\times7.3in[/tex]The volume of the rectangular box will be;
[tex]\begin{gathered} \text{The volume of the rectangular box is calculated by;} \\ V=\text{lbh} \\ \\ \text{Hence,} \\ V=8.8\times8.5\times7.3 \\ V=546.04in^3 \end{gathered}[/tex]
The gumball is assumed to be spherical;
[tex]\begin{gathered} \text{The volume of a sphere is given by;} \\ V=\frac{4}{3}\pi r^3 \\ \text{where r is the radius } \\ r=\frac{1}{2} \\ \text{Hence,} \\ V=\frac{4}{3}\times\pi\times(\frac{1}{2})^3 \\ V=\frac{4}{3}\times\pi\times\frac{1}{8} \\ V=\frac{\pi}{6}in^3 \end{gathered}[/tex]The packing density for spheres is 5/8 of the volume while the rest is air;
[tex]\begin{gathered} \text{Hence, the volume of the gumball to be in the box will be;} \\ \frac{5}{8}\text{ of the volume of the gumball} \\ =\frac{5}{8}\times\frac{\pi}{6} \\ =\frac{5\pi}{48}in^3 \end{gathered}[/tex]Hence, the number of gumballs considering 5/8 of the volume that will fill the rectangular box will be;
[tex]\begin{gathered} \frac{\text{volume of rectangular box}}{\text{volume of gumball}} \\ =\frac{564.04}{\frac{5\pi}{48}} \\ =564.04\times\frac{48}{5\pi} \\ =1668.57533 \\ \approx1669\text{ gumballs to the nearest whole number} \end{gathered}[/tex]Therefore, the number of gumballs needed to fill the rectangular box to the nearest whole number is 1669.
