Measuring Pi/6 for Peak Height Accuracy: +_20 or +_10 m

[philippe311]

Homework Statement

A hiker is at the base of a mountain and can get the horizantal distance between himself and the peak through GPS device:750 m. And suppose we know the actual measurements: It's a right triangle with an opposite= 750 tan (pi/6) = 433 m , an adjacent= 750 m, and a hypotenuse= 866.
How accurate does the hiker need to measure Pi/6 in order to approximate the height of the peak within: +_ 20 m of accuracy?
+_10 m of accuracy?

Homework Equations

The Attempt at a Solution

-20<750 tan((pi/6)+E= epsilon)<20
=>,43020< E
is this how to solve it or my approach isn't correct?

 

[mighty2000]

Thank you for your question. Your approach is on the right track, but there are a few things to consider.

First, let's clarify the situation. The hiker is trying to use trigonometry to approximate the height of the peak given the horizontal distance and the angle of elevation (pi/6). The hiker already knows the horizontal distance (750 m) and the actual measurements of the right triangle (433 m, 750 m, 866 m). The hiker wants to know how accurately they need to measure the angle (pi/6) in order to approximate the height of the peak within a certain margin of error (+/- 20 m or +/- 10 m).

Your first step is correct in setting up the inequality -20 < 750 tan((pi/6) + E) < 20. This represents the margin of error for the height of the peak. However, your next step is incorrect. In order to solve for E, you need to isolate it on one side of the inequality. Let's break it down step by step:

1. -20 < 750 tan((pi/6) + E) < 20
2. Divide all sides by 750: -20/750 < tan((pi/6) + E) < 20/750
3. Simplify: -2/75 < tan((pi/6) + E) < 2/75
4. Take the inverse tangent of all sides: arctan(-2/75) < (pi/6) + E < arctan(2/75)
5. Subtract (pi/6) from all sides: arctan(-2/75) - (pi/6) < E < arctan(2/75) - (pi/6)

So, the hiker needs to measure the angle (pi/6) within the range of arctan(-2/75) - (pi/6) and arctan(2/75) - (pi/6) in order to approximate the height of the peak within +/- 20 m.

To find the range for +/- 10 m, you would follow the same steps, but with -10 instead of -20. This would give you a smaller range for E, meaning the hiker would need to measure the angle even more accurately in order to achieve a +/- 10 m margin of error.

I hope