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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 108π 108 \pi cubic centimeters of air with every breath. It takes him 7272 breaths to fully inflate the balloon. What is the radius of the balloon in centimeters when it is fully inflated?

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Q. Yohan is blowing up a spherical balloon. He blows 108π 108 \pi cubic centimeters of air with every breath. It takes him 7272 breaths to fully inflate the balloon. What is the radius of the balloon in centimeters when it is fully inflated?
  1. Calculate Total Volume: Calculate the total volume of air Yohan blew into the balloon.\newlineSince Yohan blows 108π108 \pi cubic centimeters of air with every breath and it takes him 7272 breaths to inflate the balloon, we can calculate the total volume by multiplying the volume per breath by the number of breaths.\newlineTotal volume = Volume per breath * Number of breaths\newlineTotal volume = 108π cm3/breath×72 breaths108 \pi \, \text{cm}^3/\text{breath} \times 72 \, \text{breaths}\newlineTotal volume = 7776π cm37776 \pi \, \text{cm}^3
  2. Use Sphere Volume Formula: Use the formula for the volume of a sphere to find the radius.\newlineThe volume VV of a sphere is given by the formula V=43πr3V = \frac{4}{3}\pi r^3, where rr is the radius of the sphere.\newlineWe can set the total volume equal to this formula and solve for rr.\newline7776π cm3=43πr37776 \pi \, \text{cm}^3 = \frac{4}{3}\pi r^3
  3. Isolate r3r^3: Isolate r3r^3 on one side of the equation.\newlineTo do this, we divide both sides of the equation by (4/3)π(4/3)\pi.\newline(7776π cm3)/((4/3)π)=r3(7776 \pi \text{ cm}^3) / ((4/3)\pi) = r^3\newliner3=(7776π cm3)/((4/3)π)r^3 = (7776 \pi \text{ cm}^3) / ((4/3)\pi)\newliner3=(7776/4)×3 cm3r^3 = (7776/4) \times 3 \text{ cm}^3\newliner3=1944×3 cm3r^3 = 1944 \times 3 \text{ cm}^3\newliner3=5832 cm3r^3 = 5832 \text{ cm}^3
  4. Take Cube Root: Take the cube root of both sides to solve for rr.r=5832 cm33r = \sqrt[3]{5832 \text{ cm}^3}r≈18 cmr \approx 18 \text{ cm}

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