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Integration

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Question 317

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=34​πr3
a.

Find dVdr\frac{dV}{dr}drdV​ giving your answer in its simplest form.

[2]
b.

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≥0

Given 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.

[4]
c.

Find the radius of the balloon when t=12.5t = 12.5t=12.5, giving your answer in cm to 3 significant figures.

[3]
d.

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.

[3]

Integration Questions

  1. A Level
  2. /Maths
  3. /Integration

Practise Edexcel A Level Maths Integration with exam-style questions for A Level Maths. 437 questions covering 11.1 Integrating Standard Functions, 11.2 Integrating f(ax + b), 11.3 Using Trigonometric Identities, 11.4 Reverse Chain Rule, 11.5 Integration by Substitution, 11.6 Integration by Parts, 11.7 Partial Fractions, 11.8 Finding Areas, 11.9 The Trapezium Rule, 11.10 Solving Differential Equations, and 11.11 Modelling with Differential Equations, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.

Question bank