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Chapter 1: Problem 4

A spherical balloon is partially blown up and its surface area is measured. More air is then added, increasing the volume of the balloon. If the surface area of the balloon expands by a factor of 2.0 during this procedure, by what factor does the radius of the balloon change? (tutorial: car on curve)

Short Answer

Expert verified
Answer: The radius of the balloon expands by a factor of sqrt(2), or approximately 1.414.

Step by step solution

01

Express the surface area of a sphere in terms of its radius

The formula for the surface area (A) of a sphere with radius r is given by the equation: A = 4蟺r^2 Here, r is the initial radius of the spherical balloon before more air is added.
02

Analyze the change in surface area

The problem states that the surface area expands by a factor of 2.0. Therefore, the new surface area (A') can be expressed as: A' = 2A = 2(4蟺r^2) = 8蟺r^2
03

Express the new radius in terms of the old radius

Let r' be the new radius of the balloon. The surface area formula for the new radius can be expressed as: A' = 4蟺(r')^2 Now, we have two expressions for A': A' = 8蟺r^2 A' = 4蟺(r')^2
04

Set up the equation to find the ratio of the new radius to the old radius

Since both expressions represent the new surface area, we can set them equal to each other: 8蟺r^2 = 4蟺(r')^2
05

Solve for the ratio of the new radius to the old radius

Divide both sides of the equation by 4蟺r^2: 2 = (r')^2 / r^2 To find the factor by which the radius changes, take the square root of both sides of the equation: sqrt(2) = r' / r Now we have the factor by which the radius of the balloon changes: Factor = r' / r = sqrt(2) The radius of the balloon changes by a factor of sqrt(2) or approximately 1.414 when the surface area expands by a factor of 2.0.

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