The volume V cm3V\text{ cm}^3V cm3 of a spherical weather balloon with radius r cmr\text{ cm}r cm is given by the formula
V=43πr3 V = \frac{4}{3}\pi r^3 V=34πr3Find dVdr\frac{dV}{dr}drdV giving your answer in its simplest form.
At time ttt seconds, gas is pumped into the balloon such that the volume is increasing according to the differential equation
dVdt=1800(2t+5)2,t≥0 \frac{dV}{dt} = \frac{1800}{(2t + 5)^2}, \quad t \ge 0 dtdV=(2t+5)21800,t≥0Given that the balloon is empty at t=0t = 0t=0:
(i) Solve this differential equation to show that
V=360t2t+5 V = \frac{360t}{2t + 5} V=2t+5360t(ii) Hence determine the maximum theoretical volume of the balloon.
Find the radius of the balloon when t=12.5t = 12.5t=12.5, giving your answer in cm to 3 significant figures.
Calculate the rate of increase of the radius of the balloon when t=12.5t = 12.5t=12.5. Give your answer to 2 significant figures, show your working, and include the units of your answer.
