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3.6 Differentiation (A-level only)

3.6 Differentiation (A-level only)

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Question 249

A circular safety valve with radius 8 cm is designed to open gradually. The cross-section of the opening is a segment of a circle with centre O O\,O and radius rrr. The angle subtended by the segment at the centre is θ \theta\,θ radians.

Given that:

  • θ \theta\,θ is increasing at a constant rate of 0.375 radians per second
  • the area of the opening is A A\,A cm2^22
a.

Show that

dAdθ=K(1−cos⁡θ) \frac{dA}{d\theta} = K(1 - \cos \theta) dθdA​=K(1−cosθ)

where K K\,K is a constant to be found.

[4]
b.

Find, in cm2 ^2\,2 s−1^{-1}−1, the rate of increase of the area of the valve opening when θ=π4\displaystyle \theta = \frac{\pi}{4}θ=4π​. Give your answer in the form a+b2a + b\sqrt{2}a+b2​, where a a\,a and b b\,b are integers.

[4]
Markscheme

3.6 Differentiation (A-level only) Questions

  1. A Level
  2. /Maths
  3. /3.6 Differentiation (A-level only)

333 exam-style questions on WJEC A Level Maths 3.6 Differentiation (A-level only), covering 3.6.1 Differentiation (A-level only), 3.6.2 Differentiation (A-level only), 3.6.3 Differentiation (A-level only), 3.6.4 Differentiation (A-level only), 3.6.5 Differentiation (A-level only), 3.6.6 Differentiation (A-level only), 3.6.7 Differentiation (A-level only), and 3.6 Differentiation (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.

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