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littlekori's avatar

Can you solve this?

Asked by littlekori (676points) March 8th, 2011

A hawk sitting on a tree branch spots a mouse on the ground 15 feet away from the base of the tree. The hawk swoops down towards the mouse at an angle of 30 degrees. What is the distance from the tree branch to the mouse?

This is not homework or an assignment it is part of a practice test that I am choosing to take and I am unsure of how to solve this problem. Can anyone help me? Please!


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17 Answers

blueiiznh's avatar

Simple trig functions.
Draw it out and apply it with some concepts

tedd's avatar

To solve this you are assuming that the tree and the ground are forming a right triangle with the distance from branch to the mouse being the hypotenuse. From there its a basic geometry question. If the height of the tree is one side of the triangle (we’ll say H), and the distance from the tree bottom to the mouse is another side (we’ll say D), and your angle the bird is taking to the mouse is 30 degrees (we’ll call it AB)..... You have a 30–60-90 triangle, so your sides are at a ratio of 1–2-sqrt(3) (in D-Hypot.-H) ... so Your D=15, multiply it by 2 to get your hypotenuse… so 30.

edit: this may help

Mariah's avatar

@tedd‘s math is spot-on but I’m interpreting “the hawk swoops down towards the mouse at an angle of 30 degrees” differently that he did, so I’m getting a different answer.

Definitely draw a picture for this kind of problem. As @tedd said this is a right triangle situation with the tree forming one leg, the line on the ground between the mouse and the tree forming another leg (15), and the hypotenuse being the distance between the mouse and the hawk, which is what you’re solving for (x). Your right angle is between the tree and the ground. But where my interpretation differs from @tedd‘s is that I think the 30 degree angle is the one between the ground and the hypotenuse, not the one between the tree and the hypotenuse. I might be wrong, these sorts of word problems are tricky like that, and I feel like it’s worded somewhat ambiguously. This make the tree your shorter leg, because it is opposite the smallest angle. You can use the ratios – in a 30–60-90 right triangle, the sides opposite the 30–60-90 degree angles are at a ratio of 1-sqrt(3)-2 respectively. If you draw a diagram, it will be clear that your 15 unit long side (the ground) is opposite the 60 degree angle, and your x is opposite the 90 degree angle. So the ratio sqrt(3)/2 = 15/x – then just solve for x. My answer is 30/sqrt(3).

tedd's avatar

@Mariah Yah I took the angle as being the one between the branch and the tree (the 30 degree angle). I could have read that wrong though. If I did then Mariah’s answer would be the right one.

The picture definitely helps in this question.

Response moderated (Unhelpful)
littlekori's avatar

@tedd and @Mariah there is a picture with the problem and the 30 degrees is between the tree and the hypotenuse.

Mariah's avatar

@littlekori Ah, good that they provide a picture – I stand corrected! @tedd‘s answer should be right, then.

earthyearth's avatar

just do sin30 = 15/x = 15/sin30

earthyearth's avatar

the answer is 30 :)

littlekori's avatar

@earthyearth i think you are right because that is one of the choices on the practice test. But I am unsure of how you got 30. Like I know you use sin and what not but I don’t exactly how to do that.

earthyearth's avatar

@Mariah your answer is definitely wrong = =

earthyearth's avatar

do you know SOH CAH TOA? sine=opposite/hypotenuse – Cosine = adjacent/hypotenuse – Tangent = Opposite/adjacent

you need to know this, its the basic of trigonometry, when you draw the triangle out just observe the angle that is given and which sides is it related to,, in this case the given side is obviously opposite to the angle and you’re trying to find the hypotenuse which means sine is the “concept” you’re working with. Then just plug in the numbers Sin(theta) = opposite / hypotenuse which means sin(30) = 15/hypotenuse then just plug it in the calculation

hypotenuse in this case is the distance from the hawk to the mouse :)

earthyearth's avatar

@Mariah if you do the math 3/sqrt(3) is around 17 which is fsdkdspfopfwepo

gasman's avatar

In geometry & trig there are really only 2 kinds of right triangles you have to know—45–45-90 and 30–60-90. The latter type applies here, whose property is that the short leg measures half its hypotenuse. This translates directly into hawk-mouse distance equals twice mouse-tree distance. People freak out over math even when the problems are simple. Now is the mouse were running…

Btw, the hawk’s height off the ground will be 30 feet times the sine of 60 degrees, which is half the square root of 3, about .866. The only other decimal worth memorizing is .707, the sine of 45 degrees, which is half the square root of 2.

gasman's avatar

There’s an ambiguity in the stated problem: The hawk swoops down towards the mouse at an angle of… I took this to mean angle measured with the tree (like angle of incidence in physics) but it could be angle measured with the horizon. Is the hawk’s flight steeper or shallower than 45 degrees? Mariah’s answer of 15/sqrt(3) = 17.3 feet is correct (GA) for the shallow flight.

I think only the OP can settle this.

Mariah's avatar

Given what I know now from the OP coming back and telling us that there’s a picture with the problem that shows the 30 degree angle being between the tree and the hypotenuse, I agree that the answer is 30.

This can also be solved with trigonometric functions (sin, cos, tan) but I prefer the ratios when they can be used (30–60-90 and 45–45-90 triangles) because they give you an exact answer if there’s a radical involved, while sin punched into a calculator will give you an irrational decimal.
To do this with trig functions, take note of the locations of all the information you know and are solving for (draw a picture when there isn’t one given). You know the one angle (30 degrees) and one leg (15, opposite the 30 degree angle) and have one unknown (x, the hypotenuse). Since you’re dealing with the opposite leg and the hypotenuse, you use sine rather than cosine or tangent, because as @earthyearth said, sin = opposite/hypotenuse. Some people use that SOHCAHTOA mnemonic she mentioned. So sin(30) = 15/x. Punch sin(30) into your calculator and you’ll find it equals ½, so x=30.

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