October 2020 Paper 1 Q8

OCR MEICurrent spec7 marksModellingNumerical Methods

8 Fig. 8.1 shows the cross-section of a straight driveway 4 m wide made from tarmac.

Fig. 8.1: cross-section of the driveway, a low symmetric hump 4 m wide
Fig. 8.1

The height \(h\) m of the cross-section at a displacement \(x\) m from the middle is modelled by \(h = \dfrac{0.2}{1+x^2}\) for \(-2 \leqslant x \leqslant 2\).

A lower bound of \(0.3615\,\mathrm{m^2}\) is found for the area of the cross-section using rectangles as shown in Fig. 8.2.

Fig. 8.2: the curve h = 0.2/(1 + x^2) from x = -2 to 2 with eight rectangles of width 0.5 lying under the curve
Fig. 8.2
(a) Use a similar method to find an upper bound for the area of the cross-section. [3]
(b) Use the trapezium rule with 4 strips to estimate \(\displaystyle\int_0^2 \frac{0.2}{1+x^2}\,\mathrm{d}x\). [2]
(c) The driveway is 10 m long. Use your answer in part (b) to find an estimate of the volume of tarmac needed to make the driveway. [2]
‹ Previous50 of 52 in ModellingNext ›