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At what distance does a tree 24 ft tall subtend an angle of 10'?

this is what I got, given formula: d = rθ

θ = 1/60 degree * 3.14/180 degree = 2.9 x 10^-4 or 0.00029

d = rθ = (24)(.00029) = 0.00696 mile?

here is what i got, but my answer is not correct, the correct answer for the problem is: 1.6 mile

but I have no idea how to get 1.6 mi

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There are a few things wrong here. One, you should be using $\tan$ rather than the approximation $\tan x \approx x$. Two, You are trying to compute $r$, not $d$, and three, you need to use consistent units (you used 24' above, and quoted an answer in miles). I believe the answer is more like $r \approx 1.563$ miles. –  copper.hat Mar 5 '13 at 0:09
    
alright, thank guys –  Rex Rau Mar 5 '13 at 0:56

1 Answer 1

up vote 0 down vote accepted

In the formula that you are quoting, the height of the tree is the $d$, and the $r$ is the required distance (the "radius"). That is the opposite of the interpretation taken in the posted solution.

Since $d=r\theta$, from algebra we get that $r=\dfrac{d}{\theta}$.

First we calculate $\theta$. The angle is $10'$, that is, $10$ minutes. So $\theta$ is $\frac{10}{60}$ of a degree. In radians, that is $\dfrac{10}{60}\cdot\dfrac{\pi}{180}$. If at this stage you want to use a calculator (myself, I would wait), we get that $\theta\approx 0.0029089$. Note this is $10$ times as large as the number you computed, because the angle is $10'$, not $1'$.

Now divide $d$, that is, $24$, by $\theta$. We get that $r\approx 8250.5922$.

But recall that we are working in feet, since the height of the tree was measured in feet. We must convert to miles. To convert, we divide by $5280$, since there are $5280$ feet in a mile.

We get approximately $1.5626$. Round appropriately.

Remark: The formula $d=r\theta$ relates the length $d$ of a circular arc, in a circle with radius $r$, to the angle $\theta$ subtended by the arc an eye at the centre of the circle.

If we are on level ground, at distance $r$ from the base of a tree growing straight up, then the actual relationship is $d=r\tan\theta$.

For "small" angles $\theta$, like ours, $\tan\theta$ is very close to $\theta$, if $\theta$ is measured in radians. In our case, to the display accuracy of my cheap calculator ($7$ decimal places) they are equal. For larger angles, the difference can be significant.

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oh, ok, so i read wrong to the problem =.=, one more question, do distance always equal to radius? and arc length always equal to height? sometime, i have no idea on which is arc length and raidus, and thank for your answer :D –  Rex Rau Mar 5 '13 at 0:37
    
A nice rule of thumb (from early aircraft navigation) is the one-in-sixty rule, which is that a degree deviation corresponds to about a one mile error after flying for sixty miles. ($\frac{1}{\tan 1°} \approx 57.3 $.) –  copper.hat Mar 5 '13 at 3:00

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