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Chapter 1: Problem 20

What is the ratio of the volume of a cube of side \(r\) to that of a sphere of radius \(r\) ? Does your answer depend on the particular value of \(r ?\)

Short Answer

Expert verified
Answer: The ratio of the volume of a cube to the volume of a sphere when both have the same length r is \(\frac{3}{4\pi}\). This ratio does not depend on the particular value of r.

Step by step solution

01

Find the volume of the cube

First, we need to find the formula for the volume of a cube with side length \(r\). The volume of a cube is given by: Volume of cube = Side × Side × Side So, the volume of our cube with side length \(r\) is: \(V_{cube} = r^3\)
02

Find the volume of the sphere

Next, we need to find the formula for the volume of a sphere with radius \(r\). The volume of a sphere is given by: Volume of sphere = \(\frac{4}{3}\)π × Radius³ So, the volume of our sphere with radius \(r\) is: \(V_{sphere} = \frac{4}{3}\pi r^3\)
03

Find the ratio between the volumes of the cube and the sphere

Now, we have the volumes of both shapes: \(V_{cube} = r^3\) and \(V_{sphere} = \frac{4}{3}\pi r^3\). To find the ratio of their volumes, we will divide the volume of the cube by the volume of the sphere: Ratio = \(\frac{V_{cube}}{V_{sphere}}\) Plug in the formulas for the volumes: Ratio = \(\frac{r^3}{\frac{4}{3}\pi r^3}\)
04

Simplify the ratio expression

To simplify the ratio, we can cancel the \(r^3\) term in the numerator and the denominator: Ratio = \(\frac{r^3}{\frac{4}{3}\pi r^3} = \frac{1}{\frac{4}{3}\pi}\) Now, we can further simplify the expression: Ratio = \(\frac{3}{4\pi}\)
05

Check for dependence on \(r\)

Looking at our final ratio expression, \(\frac{3}{4\pi}\), we see that it does not contain the variable \(r\). This means that the ratio of the volume of a cube to the volume of a sphere with side length or radius \(r\) does not depend on the particular value of \(r\).

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