Question

here is a three - dimensional object shaped like a ball.
diagram shows a rectangular box containing six spheres.
the diagram also shows how the same six spheres could be packaged into a cylindrical tube.
the spheres fit tightly into the box and tube, as shown.
all spheres in both the box and the tube are exactly the same size.
the volume of each sphere is 150 cm³.
show clear working to decide which container, rectangular or cylindrical, has the greater empty
space by calculating the percentage of empty space inside each container.

Explanation:

Step1: Find the radius of the sphere

The volume formula of a sphere is \(V = \frac{4}{3}\pi r^{3}\). Given \(V = 150\mathrm{cm}^{3}\), we have \(\frac{4}{3}\pi r^{3}=150\), then \(r^{3}=\frac{150\times3}{4\pi}=\frac{450}{4\pi}\approx\frac{450}{4\times3.14}\approx35.8\), and \(r\approx3.3\mathrm{cm}\). The diameter \(d = 2r\approx6.6\mathrm{cm}\).

Step2: Calculate the volume of the rectangular box

In the rectangular box, assume there are 3 spheres in length and 2 spheres in width (since there are 6 spheres). The length \(l=3d\approx3\times6.6 = 19.8\mathrm{cm}\), the width \(w = 2d\approx2\times6.6=13.2\mathrm{cm}\), and the height \(h=d\approx6.6\mathrm{cm}\). The volume of the rectangular box \(V_{box}=l\times w\times h\approx19.8\times13.2\times6.6 = 19.8\times87.12=1725\mathrm{cm}^{3}\). The total volume of the 6 - sphere is \(V_{spheres}=6\times150 = 900\mathrm{cm}^{3}\). The percentage of empty space in the box \(P_{box}=\frac{V_{box}-V_{spheres}}{V_{box}}\times100\%=\frac{1725 - 900}{1725}\times100\%=\frac{825}{1725}\times100\%\approx47.8\%\).

Step3: Calculate the volume of the cylindrical tube

In the cylindrical tube, the height \(H = 6d\approx6\times6.6=39.6\mathrm{cm}\), and the radius of the base \(R = d/2\approx3.3\mathrm{cm}\). The volume formula of a cylinder is \(V_{cylinder}=\pi R^{2}H\approx3.14\times3.3^{2}\times39.6\approx3.14\times10.89\times39.6\approx3.14\times431.244 = 1354\mathrm{cm}^{3}\). The percentage of empty space in the cylinder \(P_{cylinder}=\frac{V_{cylinder}-V_{spheres}}{V_{cylinder}}\times100\%=\frac{1354 - 900}{1354}\times100\%=\frac{454}{1354}\times100\%\approx33.5\%\).

Answer:

The rectangular container has a greater percentage of empty space (\(47.8\%\) for the rectangular box and \(33.5\%\) for the cylindrical tube).