Higher trigonometry?

Question

A mountain rescue team has to send a rescue line across to a climber stranded on a rock face. They cannot reach the base of the face because of obstacles but observe that the angle of elevation of the stranded climber is 26.6degrees and if they move 10metres nearer, the angle of elevation becomes 30.3degrees. How high up the rock face is the climber and how long must the rescue line be in order to reach her?

How do I work this out? I've tried sketching it but I just can't figure out what it should look like. Please help me out.

Answer 1

Draw triangle ABC
AB is vertical = h m
CB is horizontal = x m
DC is horizontal = 10 m
/_ADC = 26.6°
/_ACB = 30.3°

tan 30.3° = h / x
tan 26.6° = h / (x + 10)

x tan 30.3° = (x + 10) tan 26.6°
x (tan 30.3° - tan 26.6°) = 10 tan 26.6°
x = 39.3 m

tan 30.3° = h / 39.3
h = 39.3 tan 30.3°
h = 23 m (to nearest m)

length of line = AC
AC² = 39.3² + 23²
AC = 46 m (to nearest m)
length of line from nearest point = 46 m

Answer 2

No calculator so you can work out the algebra...but the stranded climber is at "y" meters above the horizontal and his/her position does not change.

You may let the distance from the rescuer to the rock face drawn horizontal is "x"

Then the tangent of 26.6 deg = y/x (or opposite over adjacent)

When you move 10 meters closer to the stranded climber (or "x - 10" meters), the angle of elevation is 30.3 deg.

Again, you have a tangent relationship: tangent of 30.3 deg = y (which is a constant)/(x - 10)

You may then divide one expression by the other. The y values will cancel and you may solve for x. Once you solve for x you may then solve for y (the height up the rock face the climber is).

tan26.6 = y/x div by tan 30.3 = y/(x - 10); then

tan26.6/tan30.3 = (x - 10)/x

Hope this gives you the correct solution and Happy Thanksgiving!