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1.9 Calculus

1.9 Calculus

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Question 507
i.

A spherical drop of industrial lubricant is being injected into a precision-engineered cavity. The volume, VVV, of the drop is increasing at a constant rate of 180π mm3 s−1180\pi \text{ mm}^3\text{ s}^{-1}180π mm3 s−1. Calculate the rate of increase of the radius, rrr, of the drop in mm s−1 \text{mm s}^{-1}mm s−1 at the moment when the radius is exactly 3 mm3 \text{ mm}3 mm. [The volume VVV of a sphere of radius rrr is given by V=43πr3V = \frac{4}{3}\pi r^3V=34​πr3]

[5]
ii.

The depth of sediment, y metresy \text{ metres}y metres, settling at the bottom of an industrial filtration tank is monitored. The rate of change of the depth of the sediment is modeled by the differential equation

dydt=ky2 \frac{\text{d}y}{\text{d}t} = \frac{k}{y^2} dtdy​=y2k​

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

  • at the start of monitoring (t=0t = 0t=0), the sediment depth was 2 metres2 \text{ metres}2 metres.
  • after 555 hours of monitoring, the sediment depth had reached 4 metres4 \text{ metres}4 metres.
  • after TTT hours of monitoring, the sediment depth reached 6 metres6 \text{ metres}6 metres.

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

[8]
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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