Question

Manuel bought a balloon (that is a perfect sphere) with a radius of 2 cm. He wanted his balloon to be bigger, so he blew 2 big breaths of air into the balloon. Each big breath increased the balloon's radius by 1 cm. What is the ratio of the current volume of the balloon to the original volume of the balloon?
Options

(A) 1/8

(B) 1/4

(C) 4

(D) 8

Answer 1

well, the balloon original was of radius = 2 = r, he then put it 1 cm extra twice, so it went up to 4 = r.  The balloon original had a radius of 2 and then it went up to 4.

$\bf \textit{volume of a sphere, \underline{originally}}\\\\ V=\cfrac{4\pi r^3}{3}~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ r = 2 \end{cases}\implies V=\cfrac{4\pi (2)^3}{3}\implies V=\cfrac{32\pi }{3} \\\\\\ \textit{volume of a sphere, \underline{later on}}\\\\ V=\cfrac{4\pi r^3}{3}~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ r = 4 \end{cases}\implies V=\cfrac{4\pi (4)^3}{3}\implies V=\cfrac{256\pi }{3} \\\\[-0.35em] ~\dotfill$

$\bf \cfrac{\textit{later on}}{\textit{originally}}\qquad \qquad \cfrac{\frac{256\pi }{3}}{~~ \frac{32\pi }{3}~~}\implies \cfrac{256\pi }{3}\cdot \cfrac{3}{32\pi }\implies 8$