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Yohan is blowing up a spherical balloon. He blows
108 pi cubic centimeters of air with every breath. It takes him 72 breaths to fully inflate the balloon. What is the radius of the balloon in centimeters when it is fully inflated?

Yohan is blowing up a spherical balloon. He blows $108 \pi$ cubic centimeters of air with every breath. It takes him $72$ breaths to fully inflate the balloon. What is the radius of the balloon in centimeters when it is fully inflated?

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Solve proportions: word problems

Full solution

Q. Yohan is blowing up a spherical balloon. He blows

108 \pi

cubic centimeters of air with every breath. It takes him

72

breaths to fully inflate the balloon. What is the radius of the balloon in centimeters when it is fully inflated?

Calculate Total Volume: Calculate the total volume of air Yohan blew into the balloon. $\newline$ Since Yohan blows $108 \pi$ cubic centimeters of air with every breath and it takes him $72$ breaths to inflate the balloon, we can calculate the total volume by multiplying the volume per breath by the number of breaths. $\newline$ Total volume = Volume per breath * Number of breaths $\newline$ Total volume = $108 \pi \, \text{cm}^3/\text{breath} \times 72 \, \text{breaths}$ $\newline$ Total volume = $7776 \pi \, \text{cm}^3$
Use Sphere Volume Formula: Use the formula for the volume of a sphere to find the radius. $\newline$ The volume $V$ of a sphere is given by the formula $V = \frac{4}{3}\pi r^3$ , where $r$ is the radius of the sphere. $\newline$ We can set the total volume equal to this formula and solve for $r$ . $\newline$ $7776 \pi \, \text{cm}^3 = \frac{4}{3}\pi r^3$
Isolate $r^3$ : Isolate $r^3$ on one side of the equation. $\newline$ To do this, we divide both sides of the equation by $(4/3)\pi$ . $\newline$ $(7776 \pi \text{ cm}^3) / ((4/3)\pi) = r^3$ $\newline$ $r^3 = (7776 \pi \text{ cm}^3) / ((4/3)\pi)$ $\newline$ $r^3 = (7776/4) \times 3 \text{ cm}^3$ $\newline$ $r^3 = 1944 \times 3 \text{ cm}^3$ $\newline$ $r^3 = 5832 \text{ cm}^3$
Take Cube Root: Take the cube root of both sides to solve for $r$ . $r = \sqrt[3]{5832 \text{ cm}^3}$ $r \approx 18 \text{ cm}$

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