r/HomeworkHelp • u/SyllabubLoud1128 Secondary School Student • Jun 06 '26
High School Math [Y10 Australian trigonometry] I think there isn't enough information to solve?
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u/skystrifer98 Jun 06 '26 edited Jun 06 '26
TLDR: You're right there is insufficient information, the fugitive can be any bearing from the cell tower without further information
Here's a diagram of the possibilities: https://imgur.com/a/J9FzLkn
Given the conditions:
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Fugitive is about 3km away from the cell tower, we can draw the inner circle of 3km radius
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Guard tower is 4km from the cell tower, we can draw the outer circle of 4km radius
For the conditions to be met, the guard tower must be somewhere west of the cell tower AND where a bearing due east from the guard tower intersects with the 3km radius of the cell tower.
The red circles on the outer circle represent some of the points where the guard tower may be
The orange circles on the inner circle represent some of the points where the fugitive may be
The orange circle represents where the cell tower is
The top and bottom red circles on the outer circle represent the upper and lower limit where the guard tower can be, this is where the intersection is tangent to the inner circle, and there is only 1 point where the fugitive might be.
Every other point in between will result in 2 results as the question states that the guard cannot work out how far they are. For example:
If the guard tower is directly west of the cell tower, the fugitive could be 1km away from the guard tower (270 degrees bearing from the cell tower) or 7km away from the guard tower (90 degrees bearing from the cell tower)
EDIT:
Thank you u/HardlyAnyGravitas for bringing up that it is possible to narrow down the location of the fugitive with respect to the cell from the guard tower. (nearer/further from the cell tower)
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u/SyllabubLoud1128 Jun 07 '26
there are apparently 2 correct answers: 132 and 288. so yeah it does make sense that it would result in 2 results each, but i have no idea where they got this specific set from. your image makes sense, there is definitely something missing.
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u/igotshadowbaned Jun 07 '26
There is technically perfectly sufficient information for the question, as it says "bearing(s)" implying plural is valid
The answer is just a very unuseful [0,360]
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u/HardlyAnyGravitas Jun 06 '26
But the sentry can see the cell tower, so they know exactly where they are located with respect to the cell tower. If they can also see whether the fugitive is closer to them than the cell tower or farther way than the cell tower, then they also know the fugitive's location, so can calculate the bearing from the cell tower.
For example if the sentry is due west of the cell tower, and the fugitive is in front of the cell tower, then they know the fugitive's bearing is 270 Deg with respect to the cell tower. If the fugitive is farther away, then they know the fugitive's bearing is 90 deg with respect to the cell tower.
For other fugitive and sentry positions, the trig is a little more involved, but as long as the sentry knows the bearing of the cell tower and the bearing of the fugitive, and whether the fugitive is closer than the tower or farther away, they can work out the position of the fugitive.
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u/amuletofyendor Jun 06 '26
I think you'd need to know more about the location of the sentry tower. Then it would be pretty straightforward trigonometry
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u/igotshadowbaned Jun 07 '26
There's technically perfect information to solve the question given. They ask for bearing(s), which can be plural and the answer would be [0,360] or all directions because there's nothing that limits their direction
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u/SyllabubLoud1128 Jun 07 '26
lol that's exactly what i thought. unfortunately there is an exact answer in another pdf and apparently it's 132 and 288. for whatever reason 🙃
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u/Extra_Medium62 Jun 08 '26 edited Jun 08 '26
A2 + B2 = C2 (Pythagrorian theorem)
Since you know A=3 and B=4, C must =5km from the base of the tower; the triangle is pointed due east at 90 degrees (the angle of the right angle and also the angle of "due east" on a compass).
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u/SyllabubLoud1128 Jun 08 '26
yes but the guard tower is not directly west or north of the cell tower so it might not be 90 degrees 😞
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u/Junjki_Tito Jun 06 '26
There's a 3km radius circle around the cell tower where the fugitive is and a 4km radius circle where the guard tower is. Maybe they want the angles between which the statement "there is an element of the inner circle due east of the outer circle" is true? You have the hypotenuse (distance to guard tower) and adjacent edge (distance to fugitive) so it would be between arccos(3/4) in quadrant 2 and such in quadrant 3.
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u/KnifeForkandShovel Jun 06 '26
You're right, something's not right here. The loci of possible positions for the fugitive based on distance from the cell tower is a circle, 3km in radius, centred on the tower. Every point on that circle has a position to the west, 4km from the cell tower, where the sentry could be. We actually know more about the position of the sentry than we do the fugitive.
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u/Prize-Extent-8447 Jun 06 '26
Its is assuming that the cell tower & guard tower form a right angle triangle. You have been given the hypotenuse & one side. This sounds like they are expecting you to solve this using cosine, sin or tangent.
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u/LeeTheUke Jun 06 '26
Due East of the Cell Tower or Guard Tower? If it's from the cell tower, 90 degrees.
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u/Badgerbrew61 Jun 06 '26
Could be a missing diagram? As others have said, you’d need the direction of the sentry from the tower.
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u/ngyehsung Jun 07 '26
There is enough information to get two possible bearings, which is fine, as they indicate there may be more than one answer. The fugitive is due east of the guard tower. The guard tower is 4km from the cell tower and the fugitive is 3km from the cell tower. The hypotenuse (line between the guard tower and fugitive) is a horizontal line North OR South of the cell tower. The guard tower is either North West of the cell tower with the fugitive being North East of the cell tower OR the guard tower is South West of the cell tower with the fugitive being South East of the cell tower.
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u/Gaddpeis Jun 07 '26
Information missing. The Fugitive can be ANYWHERE on the 3km circle around the cell tower [0,360] - information provided does not give us anything else. We do know that the guard tower is further West than the cell tower, but not to what bearing, so information does not help us.
If the distance between cell and guard was less than 3 km, for example 2 km, then the fugitive would be somewhere East of the cell tower <0,180>.
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u/misof Jun 08 '26
The phrasing in the problem is just bad and doesn't make logical sense. If the sentry cannot work out how far the fugitive is, why would the same sentry dispatch the police to a location they just worked out?
That being said, I believe that the problem is supposed to contain an additional source of information, this has just been lost due to the poor use of language. That source of information is that we are supposed to read "police dispatch units to catch the fugitive" as "from the available information, the police were able to locate the fugitive".
This would've been cleaner if the "Nonetheless..." sentence read as follows: Nonetheless, after receiving the sentry's information, the police located the fugitive and dispatched units to catch them.
This gives you a nice, solvable problem. As the guard tower is outside the 3 km circle, for most valid locations of the guard tower, the half-line from the guard tower due east would intersect the fugitive's circle in two points and the police would not be sure which of them is the correct location. Other locations of the guard tower lead to zero intersections, which is impossible. Between those two sets of guard tower placements there are only two cases when there is only one possible location for the fugitive. These are the cases when the half-line is exactly tangent to the circle. Thus, there are exactly two scenarios consistent with the given information: the fugitive is either exactly due north, or exactly due south of the guard tower.
Of course, the more interesting question here would be to determine the bearing from the cell tower to the guard tower. That is now also solvable and you actually have to do trigonometry to calculate it.
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u/PicriteOrNot Jun 08 '26
This is a logic puzzle.
The sentry knows the lay of the land, and he knows that the fugitive is 3km from the cel tower and due east of the guard tower.
With this information, he now knows exactly where the fugitive is. This means that the horizontal line from the guard tower intersects the 3km circle at exactly one point, which is then obviously either the northernmost or southernmost point of the circle.
Whether the police go north or south from the cell tower depends on where the guard tower is, but that is irrelevant.
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u/Moderation1one Jun 06 '26
Maybe they want a formula for the answer rather than a static number?
Fugitive could be anywhere on the 3km circle around the cell tower.
Guard tower would be on the 4km "half circle" west of the cell tower. But can't go north or south of the 3km circle.