A garden parasol is held upright by a support strut \(4.5\) m long, pegged into the lawn at \(72^{\circ}\) to the horizontal and clipped onto the pole above...
Question 1 Report
A garden parasol is held upright by a support strut \(4.5\) m long, pegged into the lawn at \(72^{\circ}\) to the horizontal and clipped onto the pole above.
- Work out how high up the pole the strut is clipped. Give your answer correct to 3 significant figures. (2)
- Work out how far the peg is from the base of the pole. Give your answer correct to 3 significant figures. (2)
Answer Details
The support strut is the hypotenuse of a right-angled triangle formed with the pole (vertical) and the ground (horizontal), with the \(72^\circ\) angle measured from the horizontal lawn.
- The height up the pole is opposite the \(72^\circ\) angle, so the sine ratio is used: \[ h = 4.5\times\sin 72^\circ = 4.5\times 0.9511... = 4.279... \] giving \(h = 4.28\) m (3 s.f.). [2 marks]
- The distance of the peg from the base of the pole is adjacent to the \(72^\circ\) angle, so the cosine ratio is used: \[ x = 4.5\times\cos 72^\circ = 4.5\times 0.3090... = 1.390... \] giving \(x = 1.39\) m (3 s.f.). [2 marks]
Because \(72^\circ\) is close to a right angle, the strut leans nearly vertically, so the height (\(4.28\) m) being much larger than the horizontal peg distance (\(1.39\) m) is exactly what is expected; using sine for the side opposite the angle and cosine for the side adjacent to it is the pattern to fix for any "angle to the horizontal" problem.
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