What percentage increase in the radius of a sphere will cause its volume to increase…

What percentage increase in the radius of a sphere will cause its volume to increase by 45%?

Explanation

Let the original volume be V with radius r.

(V = frac{4}{3}pi r^{3})

45% increased volume = 145%V. 

Let the %age increase in radius = m%r

(frac{145}{100}V = frac{4}{3}pi (frac{mr}{100})^{3})

(1.45V = (frac{4}{3}pi r^{3})(frac{m}{100})^{3})

(1.45V = V(frac{m}{100})^{3})

(implies 1.45 times 10^{6} = m^{3})

(m = sqrt[3]{1.45 times 10^{6}} = 113.2%)

(therefore text{%age increase =}  113.2 – 100 = 13.2%)

(approxeq 13%)