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Chapter 1: Problem 21

Consider a sphere of radius \(r\). What is the length of a side of a cube that has the same surface area as the sphere?

Short Answer

Expert verified
2. What is the formula for the surface area of a cube? 3. If the surface area of a sphere and cube are equal, what is the equation relating their side length and radius? 4. How is the length of a side of the cube found once the equation is set up?

Step by step solution

01

Find the surface area of the sphere

We are given the radius of the sphere (\(r\)) and the formula for the surface area of a sphere is \(4\pi r^2\). Therefore, the surface area of the sphere is: Sphere_Surface_Area \(= 4\pi r^2\)
02

Find the surface area of the cube

Let \(s\) be the length of a side of the cube. The formula for the surface area of a cube is \(6s^2\). So the surface area of the cube is: Cube_Surface_Area \(= 6s^2\)
03

Set the surface area of the sphere equal to the surface area of the cube

We need to find the length of a side of a cube (\(s\)) that has the same surface area as the sphere. So we will set the two surface areas equal to each other: \(4\pi r^2 = 6s^2\)
04

Solve for \(s\)

Divide both sides of the equation by 6 to isolate the variable \(s^2\): \(s^2 = \dfrac{4\pi r^2}{6}\) Now take the square root of both sides to find the value of \(s\): \(s = \sqrt{\dfrac{4\pi r^2}{6}}\) So the length of a side of a cube that has the same surface area as the sphere is: \(s = \sqrt{\dfrac{4\pi r^2}{6}}\)

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