Higher November 2017 Paper 2 Q16
16. Shireen is trying to find the area of a field, ABCDE.
She has taken a few measurements and written them on a sketch, as shown below.
Diagram not drawn to scale
Shireen has indicated
- \(D\hat{C}A = 90^\circ\)
- DC = 20 metres
- CA = 15 metres
- \(E\hat{A}D = 60^\circ\)
- \(B\hat{A}C = 45^\circ\)
She thinks that DA = EA and CA = BA.
Shireen knows that BC and DE are not straight lines.
Using the information Shireen has collected, calculate the area of the field.
You must state any assumption you make in deciding on your method. [9]
| Scheme | Marks | Comments |
|---|---|---|
| \((DA^2 =)\ 20^2 + 15^2\) or \(DA^2 = 625\) or \(DA = \sqrt{625}\) | M1 | |
| \(DA = 25\) (metres) | A1 | |
| Area triangle \((\tfrac12 \times 15 \times 20 =)\) 150 (m2) | B1 | |
| Area \(DAE\) in the range 327.08 to 327.4 or \(625\pi/6\) (m2) | B2 | FT ‘their derived 25’ provided ≠15 and ≠20 B1 for \(\pi \times 25^2 \times 60 \div 360\), OR B1 for an answer of 270.6 to 271 (m2) from Area DAE as a triangle, FT ‘their 25’ for DA: \(\tfrac12 \times 25 \times 25 \times \sin 60^\circ\) = 270.6(3.. m2) OR \(\tfrac12\)DE straight = \(25 \times \sin 30^\circ\) = 12.5(m) and A to midpoint DE = \(25 \times \cos 30^\circ\) = 21.65(… m), and 12.5 × 21.65(…) = 270.6(3…m2) |
| Area \(CAB\) in the range 88.3 to 88.4 or \(225\pi/8\) (m2) | B2 | B1 for \(\pi \times 15^2 \times 45 \div 360\), OR B1 for an answer of 79.5 to 80 (m2) from Area CAB as a triangle: \(\tfrac12 \times 15 \times 15 \times \sin 45^\circ\) = 79.5(4.. m2) OR \(\tfrac12\)CB straight = \(15 \times \sin 22.5^\circ\) = 5.7(4…m) and A to midpoint CB = \(15 \times \cos 22.5^\circ\) = 13.8(58… m), and 5.7(4…) × 13.8(58…) = 79.5(4…m2) |
| Total area answer in the range 565 to 566 (m2) | B1 | CAO. Not FT from area of triangles |
| Assumption, e.g. ‘(area of) sector(s) of a circle’, ‘Shireen is correct to think DA = EA’, ‘Shireen is correct in thinking AC = AB’ | E1 | Allow descriptions of a sector of a circle FT assumption ‘(area of) triangle(s)’ provided implies knowing that ‘their area(s)’ will be less than the actual area(s) |
| (9) |