Q1
Answer
A spherical bubble is formed and its radius is increasing at the constant rate of 0.2 cm per second.
  1. Find the rate at which the volume of the bubble is increasing when the radius of the bubble reaches 8 cm.
  2. Determine the rate at which the volume of the bubble is increasing when the surface area of the bubble reaches 264 cm.
  3. Calculate the rate at which the surface area of the bubble is increasing 30 seconds after the bubble was first formed.
  1. $\dfrac{256}{5}\pi$
  2. $\dfrac{64}{5}$
  3. $\dfrac{48}{5}\pi$
Q2
Answer
Given that $$\int_1^k\dfrac{(k\sqrt{3}+\sqrt{3x})^2}{kx^3}\ \mathrm{d}x = a-\sqrt{k}$$ where $a$ and $k$ are integers, find $a$ and $k$.
$k = \sqrt{2}$ and $a = \dfrac{9k + 2}{4}$