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1.7 Differentiation

1.7 Differentiation

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Question 186
i.

The volume, VVV, of a spherical weather balloon is increasing at a constant rate of 120π cm3 s−1120\pi \text{ cm}^3\text{ s}^{-1}120π cm3 s−1. Find the rate of increase of the radius, rrr, of the balloon in cm s−1\text{cm s}^{-1}cm s−1 at the instant when the radius is 6 cm6 \text{ cm}6 cm. [The volume VVV of a sphere of radius rrr is given by the formula V=43πr3V = \frac{4}{3}\pi r^3V=34​πr3]

[4]
ii.

The height of a pile of sand, h metresh \text{ metres}h metres, under a conveyor belt is monitored over time. The rate of increase in the height of the pile is modeled by the differential equation

dhdt=kh2 \frac{\text{d}h}{\text{d}t} = \frac{k}{h^2} dtdh​=h2k​

where kkk is a positive constant and ttt hours is the time after the measurement began. Given that:

  • initially (at t=0t = 0t=0), the height of the pile was 3 metres3 \text{ metres}3 metres.
  • 444 hours after monitoring began, the height of the pile was 5 metres5 \text{ metres}5 metres.
  • TTT hours after monitoring began, the height of the pile was 8 metres8 \text{ metres}8 metres.

Solve the differential equation to find the value of TTT. Give your answer to one decimal place.

[6]
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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