A triangular table top in the school canteen has two edges meeting at 58°. One of them is 1.4 m long and the top covers 0.85 m². Calculate the length of the...
Question 1 Report
A triangular table top in the school canteen has two edges meeting at 58°. One of them is 1.4 m long and the top covers 0.85 m².
- Calculate the length of the other edge meeting it at 58°. (2)
- Calculate the length of the third edge, to 3 significant figures. (2)
Answer Details
(a) The area formula \(\frac{1}{2}ab\sin C\) uses the two sides that meet at the known angle, so it can be rearranged to find the missing side.
- \(\frac{1}{2} \times 1.4 \times b \times \sin 58^\circ = 0.85\). [1]
- Multiplying both sides by 2 gives \(1.4 b \sin 58^\circ = 1.7\), so \(b = \frac{1.7}{1.4 \times 0.848048\ldots} = \frac{1.7}{1.18727\ldots} = 1.43185\ldots\), that is 1.43 m to 3 significant figures. [1]
(b) Two sides and the angle between them are now known, so the cosine rule gives the third edge.
- \(c^2 = 1.4^2 + 1.43185\ldots^2 - 2 \times 1.4 \times 1.43185\ldots \times \cos 58^\circ\). [1]
- \(= 1.96 + 2.05020\ldots - 4.00918\ldots \times 0.529919\ldots = 4.01020\ldots - 2.12454\ldots = 1.88566\ldots\), so \(c = \sqrt{1.88566\ldots} = 1.37319\ldots\), that is 1.37 m to 3 significant figures. [1]
Use the unrounded 1.43185 in part (b) rather than 1.43, since the value is squared and doubled in the formula. The third edge is shorter than either of the other two, which is consistent with the 58° angle being the smallest angle in the triangle.
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