If the radius of a sphere is increased by 50%, find the percent increase in surface area.
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 125%
The surface area of a sphere is \(4\pi r^{2}\), so it changes with the square of the radius: increasing the radius 1.5 times increases the area \(1.5^{2}\) times.
New radius \(=r+50\%\text{ of }r=1.5r\)
New surface area \(=4\pi(1.5r)^{2}=2.25\times4\pi r^{2}\), that is, 2.25 times the old area
Increase \(=2.25-1=1.25\) times the old area, that is, 125%
Shortcut: Area depends on \(r\times r\), so it is two successive increases of 50% each.
For successive increases of \(a\%\) and \(b\%\), the total increase is \(\displaystyle a+b+\frac{ab}{100}\)
\(\displaystyle 50+50+\frac{50\times50}{100}=100+25=125\%\)