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1.7 Differentiation

1.7 Differentiation

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Question 285

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]
Markscheme

1.7 Differentiation Questions

  1. A Level
  2. /Maths
  3. /1.7 Differentiation

425 exam-style questions on OCR A Level Maths 1.7 Differentiation, covering 1.7.1 Derivative as gradient of the tangent, 1.7.2 Gradient of the tangent at a point, 1.7.3 Sketching the gradient function, 1.7.4 Second derivatives, 1.7.5 Second derivative as rate of change of gradient, 1.7.6 Convex, concave and points of inflection (A-level only), 1.7.7 Differentiation from first principles for powers of x, 1.7.8 Differentiation from first principles for sin x and cos x (A-level only), 1.7.9 Differentiating x^n, 1.7.10 Differentiating e^(kx) and a^(kx) (A-level only), 1.7.11 Differentiating trigonometric functions (A-level only), 1.7.12 Derivative of ln x (A-level only), 1.7.13 Tangents and normals, 1.7.14 Stationary points, 1.7.15 Increasing and decreasing functions, 1.7.16 Points of inflection (A-level only), 1.7.17 Product and quotient rules (A-level only), 1.7.18 Chain rule (A-level only), 1.7.19 Parametric and implicit differentiation (A-level only), 1.7.20 Constructing differential equations (A-level only), and 1.7 Differentiation. Each one has a worked solution and a mark scheme showing where the marks go.

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