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1.9 Numerical Methods (A-level only)

1.9 Numerical Methods (A-level only)

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Question 63

The rate of airflow R R\,R in litres per second into a pneumatic cylinder is modeled by the function

R(t)=(12−3t)ln⁡t,1≤t≤4 R(t) = (12 - 3t) \ln t, \quad 1 \le t \le 4 R(t)=(12−3t)lnt,1≤t≤4

where t t\,t is the time in seconds after the valve opens. The total volume of air VVV (in litres) that enters the cylinder between t=1t = 1t=1 and t=4t = 4t=4 corresponds to the area of the region bounded by the curve R(t)R(t)R(t) and the ttt-axis.

a.

Use the trapezium rule with 5 ordinates to find an estimate for the total volume of air V V\,V that enters the cylinder. Give your answer correct to three significant figures.

[4]
b.

Show that the exact volume of air V V\,V is given by

48ln⁡2−24.75 48 \ln 2 - 24.75 48ln2−24.75

Fully justify your answer.

[6]
Markscheme

1.9 Numerical Methods (A-level only) Questions

  1. A Level
  2. /Maths
  3. /1.9 Numerical Methods (A-level only)

125 exam-style questions on OCR A Level Maths 1.9 Numerical Methods (A-level only), covering 1.9.1 Locating roots by sign change (A-level only), 1.9.2 Failure of sign change methods (A-level only), 1.9.3 Simple iterative methods (A-level only), 1.9.4 Newton-Raphson method (A-level only), 1.9.5 Failure of iterative methods (A-level only), 1.9.6 Numerical integration (A-level only), and 1.9.7 Numerical methods in context (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.

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