as the title said,
I have a little trouble to find which is radius and arc length
1)If a hill 2500 ft away subtend at 1.5 degree angle, how high is it?
my thinking:
it ask for the arc length(height), and radius(distance) = 2500, angle = 1.5
2)At what distance does a tree 24 ft tall subtend an angle of 10'?
my thinking:
it ask for the radius(distance), arc length(height) = 25, angle = 10'
so to the two questions above, my thinking is: distance is radius, height is arc length, and I calculated and it is correct, but while working on the question below, then I am confuse
3)An 18-in.mallard duck files overhead, subtending an angle of 3 degree 15'. how high is this duck flying?
my thinking:it ask for the height, radius = 18-in(1.5 ft), and the degree = 3 degree 15'(0.05669)
so i use the formula: s = rΘ = (1.5)(0.05669) = 0.085035ft , but it is wrong, the answer is 26ft which using the formula: r = s/Θ = 1.5/0.05669 = 26ft (rounded)
but isn't it asking for the high(arc length), why answer will end up with a radius(distance)?
so i am thinking it is true that arc length = height, and radius = distance? or both would switch base on different problems?
thank
Best Answer
Roughly speaking, in the formula $s=r\theta$, the symbol $s$ is the size (length) of the object you are observing, and $r$ is the distance from your eye to the object.
So in the mallard question, we have $s=18''$, and "how high it is flying" is $r$. Thus $r=\dfrac{s}{\theta}$. since you will want to give the answer in feet, you can do one of two things: (i) Set $s=18$, and compute $r$. The answer will be in inches, and you convert to feet by dividing by $12$; or (ii) Convert the $18''$ to feet ($1.5$) and compute.
For the hill, the distance you are from it ("$r$") is known, and you want $s$, the size of the hill. We use the formula $s=r\theta$. We know $r$, and $\theta$, so we multiply.