Question

1. a cube and a sphere both have volume 512 cubic units. which solid has a greater surface area? explain your reasoning.

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Explanation:

Step1: Find the side length of the cube

The volume formula of a cube is \(V = s^{3}\), where \(s\) is the side - length. Given \(V = 512\), then \(s^{3}=512\), so \(s=\sqrt[3]{512}=8\).
The surface - area formula of a cube is \(A_{cube}=6s^{2}\). Substitute \(s = 8\) into the formula: \(A_{cube}=6\times8^{2}=6\times64 = 384\).

Step2: Find the radius of the sphere

The volume formula of a sphere is \(V=\frac{4}{3}\pi r^{3}\). Given \(V = 512\), then \(\frac{4}{3}\pi r^{3}=512\).
First, solve for \(r^{3}\): \(r^{3}=\frac{512\times3}{4\pi}=\frac{384}{\pi}\approx\frac{384}{3.14}\approx122.29\).
Then \(r=\sqrt[3]{\frac{384}{\pi}}\approx\sqrt[3]{122.29}\approx4.96\).
The surface - area formula of a sphere is \(A_{sphere}=4\pi r^{2}\). Substitute \(r\approx4.96\) into the formula: \(A_{sphere}=4\pi\times(4.96)^{2}\approx4\times3.14\times24.60\approx309.0\).

Answer:

The cube has a greater surface area. The surface area of the cube is \(384\) square units and the surface area of the sphere is approximately \(309.0\) square units.