smc-4552-45-2-triangles

Measurement, STD2 M6 2019 HSC 22

Two right-angled triangles, `ABC` and `ADC`, are shown.

Calculate the size of angle `theta`, correct to the nearest minute. (3 marks)

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`41°4^{′}\ \ text{(nearest minute)}`

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`text(Using Pythagoras in)\ DeltaACD:`

Mean mark 51%.

`AC^2`	`= 2.5^2 + 6^2`
	`= 42.25`
`:.AC`	`= 6.5\ text(cm)`

`text(In)\ DeltaABC:`

`costheta`	`= 4.9/6.5`
`theta`	`= cos^(−1)\ 4.9/6.5`
	`= 41.075…`
	`= 41°4^{′}31^{″}`
	`= 41°5^{′}\ \ text{(nearest minute)}`

Measurement, STD2 M6 2010 HSC 24d

The base of a lighthouse, `D`, is at the top of a cliff 168 metres above sea level. The angle of depression from `D` to a boat at `C` is 28°. The boat heads towards the base of the cliff, `A`, and stops at `B`. The distance `AB` is 126 metres.

What is the angle of depression from `D` to `B`, correct to the nearest degree? (3 marks)

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How far did the boat travel from `C` to `B`, correct to the nearest metre? (2 marks)

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`53^circ`
`190\ text(m)`

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♦♦ Mean mark 31%

i.	`tan/_ADB`	`=126/168`
	` /_ADB`	`=36.8698…`
		`=36.9^circ\ \ \ \ text{(to 1 d.p)}`

`/_text(Depression)\ D\ text(to)\ B`	`=90-36.9`
	`=53.1`
	`=53^circ\ text{(nearest degree)}`

ii. `text(Find)\ CB:`

♦♦ Mean mark 31%
MARKER’S COMMENT: Solve efficiently by using right-angled trigonometry. Many students used non-right angled trig, adding to the calculations and the difficulty.

`/_ADC+28`	`=90`
`/_ADC`	`=62^circ`

`tan 62^circ`	`=(AC)/168`
`AC`	`=168xxtan 62^circ`
	`=315.962…`

`CB`	`=AC-AB`
	`=315.962…-126`
	`=189.962…`
	`=190\ text(m (nearest m))`

Measurement, STD2 M6 2012 HSC 27d

A disability ramp is to be constructed to replace steps, as shown in the diagram.

The angle of inclination for the ramp is to be 5°.

Calculate the extra distance, `d`, that the ramp will extend beyond the bottom step.

Give your answer to the nearest centimetre. (3 marks)

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`386\ text(cm)`

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`text(Let the horizontal part of the ramp) = x\ text(cm)`

♦♦ Mean mark 35%
MARKER’S COMMENT: The better responses used a diagram of a simplified version of the ramp as per the Worked Solution.

`tan5^@`	`= 39/x`
`x`	`= 39/tan5^@`
	`= 445.772…`

`text(S)text(ince)\ \ x`	`= 60 + d`
`d`	`=445.772-60`
	`=385.772\ text(cm)`
	`=386\ text(cm)\ \ text{(nearest cm)}`