Question

balloons the radius of a sphere r is given in terms of its volume v by the formula r = \\(\sqrt3{\frac{0.75v}{\pi}}\\). by how many inches has the radius of a spherical balloon increased when the amount of air in the balloon is increased from 4.5 cubic feet to 4.7 cubic feet? round your answer to the nearest hundredth. in.

Explanation:

Step1: Calculate initial radius

First, find the radius $r_1$ when $V_1 = 4.5$ cubic - feet. Substitute $V_1$ into the formula $r=\sqrt[3]{\frac{0.75V}{\pi}}$.
$r_1=\sqrt[3]{\frac{0.75\times4.5}{\pi}}=\sqrt[3]{\frac{3.375}{\pi}}$
Using $\pi\approx3.14159$, $r_1=\sqrt[3]{\frac{3.375}{3.14159}}\approx\sqrt[3]{1.07429}\approx1.024$ feet.
Since 1 foot = 12 inches, $r_1\approx1.024\times12 = 12.288$ inches.

Step2: Calculate final radius

Next, find the radius $r_2$ when $V_2 = 4.7$ cubic - feet. Substitute $V_2$ into the formula $r=\sqrt[3]{\frac{0.75V}{\pi}}$.
$r_2=\sqrt[3]{\frac{0.75\times4.7}{\pi}}=\sqrt[3]{\frac{3.525}{\pi}}$
Using $\pi\approx3.14159$, $r_2=\sqrt[3]{\frac{3.525}{3.14159}}\approx\sqrt[3]{1.1219}$ $\approx1.039$ feet.
Since 1 foot = 12 inches, $r_2\approx1.039\times12 = 12.468$ inches.

Step3: Calculate the increase in radius

Then, find the difference $\Delta r=r_2 - r_1$.
$\Delta r=12.468 - 12.288=0.18$ inches.

Answer:

$0.18$