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3 marksDifferentiation I

CAPE Pure Mathematics Unit 1 · May/June 2008 · Paper 2 · Question 6(c)(i)a)

The diagram below (not drawn to scale) represents a can in the shape of a closed cylinder with a hemisphere at one end. The can has a volume of 45π units³. Taking r units as the radius of the cylinder and h units as its height.

Show that h = 45/r² - 2r/3.

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Other parts of this question

  1. 6(a)(i)Differentiate with respect to x: x√(2x - 1).[3 marks]
  2. 6(a)(ii)Differentiate with respect to x: sin²(x³ + 4).[4 marks]
  3. 6(b)(i)Given that ∫(from 1 to 6) f(x) dx = 7, evaluate ∫(from 1 to 6) [2 - f(x)] dx.[3 marks]
  4. 6(b)(ii)Find the value of the constant k.[4 marks]
  5. 6(c)(i)b)Show that A = 5πr²/3 + 90π/r, where A units is the external surface area of the can.[3 marks]
  6. 6(c)(ii)Hence, find the value of r for which A is a minimum and the corresponding minimum value of A.[5 marks]

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