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1.9 Calculus

1.9 Calculus

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Question 400
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

1.9 Calculus Questions

  1. A Level
  2. /Maths
  3. /1.9 Calculus

825 exam-style questions on OCR (MEI) A Level Maths 1.9 Calculus, covering 1.9.1 Gradient of a curve at a point, 1.9.2 Gradient as limit of chord gradient, 1.9.3 Derivative as gradient of tangent, 1.9.4 Sketch the gradient function, 1.9.5 Differentiate y = kx^n, 1.9.6 Second derivative as rate of change of gradient, 1.9.7 Stationary points: maxima and minima, 1.9.8 Increasing and decreasing functions, 1.9.9 Tangent and normal at a point, 1.9.10 Differentiate e^kx, a^kx and ln x (A-level only), 1.9.11 Differentiate trigonometric functions (A-level only), 1.9.12 Product rule (A-level only), 1.9.13 Quotient rule (A-level only), 1.9.14 Chain rule (A-level only), 1.9.15 Rates of change with the chain rule (A-level only), 1.9.16 Implicit differentiation (A-level only), 1.9.17 Concavity and the second derivative (A-level only), 1.9.18 Points of inflection (A-level only), 1.9.19 Integration as reverse of differentiation, 1.9.20 Integrate kx^n, 1.9.21 Constant of integration, 1.9.22 Indefinite and definite integrals, 1.9.23 Area between a graph and the x-axis, 1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only), 1.9.25 Integration as the limit of a sum (A-level only), 1.9.26 Area between two curves (A-level only), 1.9.27 Integration by substitution (reverse chain rule) (A-level only), 1.9.28 Integration by substitution (other cases) (A-level only), 1.9.29 Integration by parts (A-level only), 1.9.30 Integration using partial fractions (A-level only), 1.9.31 Formulate first order differential equations (A-level only), 1.9.32 Solve first order differential equations (A-level only), and 1.9.33 Interpret solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.

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