Fun G&T question...(math whizzes)...?

Question

1. A surveying team marks off two positions P and Q. They are on level ground, on the same side of a hill and in line with the hill. The horizontal distance from P to Q is 1200m and the team measure the angle of elevation of the summit of a hill from P and Q as 21° and 71° respectively. Find the height of the hill.

My answer: Height of hill=822.1m (correct to 1. decimal place).

Also I live in Australia...if you are not familiar with centrimetres(cm) and metres (m) here are a few conversions:
1cm=0.01metre
10cm=0.1metre (m)
100cm=1metre

Thankyou for helping. If you get a different answer to me, could you please explain it (in steps). Also, could you please attach a diagram of the question either send it to my email or attach it after your answer. Thankyou for helping.

Could someone please attach a drawn picture of the situation if you get a different answer. I

OH NO...I typed in 21degrees for the 1st angle when it was meant to be 29degrees! All of your working was correct if it was 21degrees (I went through it all myself) and came up with the exact same answer. I think 822.1m is correct when the angles are 29degrees and 71degrees. Is this answer correct? I can't believe I typed in 21 degrees!

Answer 1

Let x = distance from the center of the bottom of the hill to P.
Let h = height of the hill. Then:

tan(71) = h/x
tan(21) = h/(x+1200)

h = x*tan(71)
h = (x+1200)*tan(21)

x*tan(71) = (x+1200)*tan(21)
x*tan(71) = x*tan(21) + 1200*tan(21)
x*tan(71) - x*tan(21) = 1200*tan(21)
x*(tan(71) - tan(21)) = 1200*tan(21)
x = 1200*tan(21) / (tan(71) - tan(21))
x = 182.767243
h = 182.767243 * tan(71) = 530.7946131 m

Yep...these numbers work out. The height of the hill is 530.7946131 m

Answer 2

Bill T and Science have already shown how to work out the solution correctly. You asked for a picture - The "hill" would be very steep - practically a cliff. The slope up the hill would have to be more than 71 degrees, definitely not a hill you can walk up.

The only think I would note about the previous solutions is the the horizontal measurement x is not the distance from Q to the base of the hill, but from Q to the point underground directly under the summit of the hill, at the same horizontal level as P and Q.

Answer 3

Let S be the summit. From the information given, angle SPQ = 21°, PQS = 180-71 = 109° and QSP = (180-21-109) = 50°.

Using the law of sines, QS = PQ*sin21°/sin50° = 561.379 m
The height is then
H = QSsin109° = 530.795 m

Sorry, I have no idea how to attach a picture here.

Answer 4

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Answer 5

You have two right triangles.

Let
h = height hill
x = horizontal distance to hill from Q
x + 1200 = horizontal distance to hill from P

We have:

tan 71° = h/x
tan 21° = h/(1200 + x)

x = h / tan 71°

1200 + x = h / tan 21°
x = h / tan 21° - 1200

Set the two equations equal.

x = h / tan 21° - 1200 = h / tan 71°

Multiply thru by (tan 71°)(tan 21°).

h(tan 71°) - 1200(tan 71°)(tan 21°) = h(tan 21°)

h(tan 71°) - h(tan 21°) = 1200(tan 71°)(tan 21°)

h[(tan 71°) - (tan 21°)] = 1200(tan 71°)(tan 21°)

h = 1200(tan 71°)(tan 21°) / [(tan 71°) - (tan 21°)]

h ≈ 530.79461 m