Hint
Use the stated units and precision. Use the two adjacent sides and their included angle in the cosine rule.
Worked solution
(a)
Use the two adjacent sides and their included angle in the cosine rule.
The cosine rule gives \(\displaystyle \sqrt{144 \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} - \frac{\sqrt{2} \left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right)}{2}\right) \left(- \frac{\sqrt{5}}{8} + \frac{1}{8} + \frac{\sqrt{3} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2}\right) + 144 \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{2} \left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right)}{2}\right) \left(- \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} + \frac{\sqrt{3} \left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right)}{2}\right) + 145}\), which rounds to \(14.0\) cm.
Answer: \(14.0\)
(b)
Use half ab sin C with the included angle.
Half the product of the adjacent sides and sin C rounds to \(33.6\) \(\displaystyle \mathrm{cm}^{2}\).
Answer: \(33.6\)
(c)
Keep the computed third side unrounded when forming the perimeter.
Add both supplied sides and the unrounded cosine-rule side to obtain \(31.0\) cm.
Answer: \(31.0\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternative methods and consequential reasoning. An unexplained result does not earn a method mark.
(a) · 2 marks
- 1 mark: Use the two adjacent sides and their included angle in the cosine rule.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Keep intermediate values exact until the final step.
(b) · 2 marks
- 1 mark: Use half ab sin C with the included angle.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Check the answer against the original conditions.
(c) · 2 marks
- 1 mark: Keep the computed third side unrounded when forming the perimeter.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Keep intermediate values exact until the final step.
Total: 6 marks.