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Chapter 1: Problem 3

A spherical balloon expands when it is taken from the cold outdoors to the inside of a warm house. If its surface area increases \(16.0 \%,\) by what percentage does the radius of the balloon change?

Short Answer

Expert verified
Answer: The radius of the balloon changes by 7.7%.

Step by step solution

01

Write down the given information

The surface area of the balloon increases by \(16.0 \%\). We'll represent this increase as a multiplier, \(1.16\), since we know that increasing by \(16.0 \%\) is equal to multiplying by \(1.16\).
02

Write down the formula for the surface area of a sphere

The formula for the surface area of a sphere with radius \(r\) is: \[A = 4\pi r^2\]
03

Use the given percentage increase to write the equation

Assume the initial radius of the balloon is \(r_1\) and the final radius is \(r_2\). According to the given information, the surface area of the balloon increases \(16.0 \%\). So, we can write the equation as: \[4\pi r_2^2 = 1.16(4\pi r_1^2)\]
04

Simplify the equation

Divide both sides of the equation by \(4\pi\): \[r_2^2 = 1.16r_1^2\]
05

Take square root of both sides

To find the relation between \(r_1\) and \(r_2\), take the square root of both sides of the equation: \[r_2 = \sqrt{1.16}r_1\]
06

Calculate the percentage change in radius

First, evaluate \(\sqrt{1.16}\): \[\sqrt{1.16} \approx 1.077\] Now we can find the percentage change in the radius from \(r_1\) to \(r_2\). The formula for the percentage change is: \[\text{Percentage Change} = \frac{\text{Final Value} - \text{Initial Value}}{\text{Initial Value}} \times 100\%\] Plug the values into the formula: \[\text{Percentage Change} = \frac{1.077r_1 - r_1}{r_1} \times 100\%\]
07

Simplify and find the percentage change in the radius

Simplify the percentage change expression: \[\text{Percentage Change} = (1.077 - 1) \times 100\%\] \[\text{Percentage Change} = 0.077 \times 100\%\] \[\text{Percentage Change} = 7.7\%\] Therefore, the radius of the balloon changes by \(7.7 \%\).

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