Trigonometry Dilemma!

[Ventus]

The question was Aeroplane A travels 400 km/h on a angle of elevation of 30 degrees. Aeroplane B travels 500km/h at a angle of elevation of 25 degrees. Supposing these angles just took of and there was no run up please answer the following questions.
1) What is the distance covered by both Aeroplanes if they were both flying for 30 minutes supposing there was no speed build up nor decrease.
(note) for this question I calculated the adjacent side as that was the distance COVERED and not how much it travelled. Am I correct of doing so?
2) Find the altitude of both aeroplanes after 30 minutes
(note) for this question I used my answer for part a and I did Pythagoras to find the opposite side or in this case the altitude. I did not use Sine to figure it out. Am I wrong in my calculations?
3) What is the difference of altitude of these 2 aeroplanes?
(note) When I calculated this with my previous answer I got a final answer of 5.66 km. However If I used sine and not Pythagoras I get 5.65 km. Which is correct?

*PS* it'd be great if you tried both ways. If you find any fault in my methods please tell me.
Thanks, Grade 9 student.

 

[Harry_the_cat]

Please show what you've done so we can see if you've gone wrong anywhere. We cant say if you are correct or not if we can't see what you have done or, at least. the solutions you got.

 

[Dr.Peterson]

I think at least part of your question is about interpreting the English.

1) I would take "distance covered" as "distance traveled" (in a straight line at constant speed). I have never heard of using the term to mean the distance along the ground.

2) You could either do what you did, or use the sine. I would not have calculated the horizontal distance for (1), so I would just use the sine here. If you are getting different answers the two ways, we need to see your work to find the error. If you are just asking whether your answer is correct, you'll have to tell us your answer!

3) Most likely the difference in your answers is due to rounding. Again, we need to see the intermediate results to be sure of that. The difference I get is 5.6545..., which rounds to 5.65. (It's best not to round until the end.)

 

[Otis]

... these angles just took of and there was no run up ...

Hello. That statement is an example of why the forum guidelines ask students to check posts before submitting them. Please use the Preview button.

... I calculated the adjacent side as ... distance COVERED and not ... travelled. Am I correct of doing so?

Yes, I think you're correct, yet I also agree with Dr. Peterson: distance covered means distance traveled. This is a trig exercise, so I think the intent is practice using sine and cosine. That means finding the opposite and adjacent sides of each triangle. So, part (1) ought to say "horizontal distance covered" and "each aeroplane."

... [For part (2)] I did Pythagoras to find the opposite side ... Am I wrong in my calculations?

As Harry_the_cat noted, we can't see your calculations. I think you meant to ask, "Was I wrong to use that method?"

Mechanically, your choice is not invalid (Pythagorean Theorem works with all right triangles). However, this is a trig exercise, so I've assumed your instructor expects to see practice using sine and cosine.

... [For part (3)] I got a final answer of 5.66 km. However If I used sine and not Pythagoras I get 5.65 km. Which is correct?

5.65 km is correctly rounded.

When we round a number, we introduce a little bit of error. If that error is used in another calculation and we round the result again, then the error grows. This issue is known as "round-off error".

When I use a decimal approximation from a calculator, I don't round it and re-enter it, for the next calculation. I let the calculator access the full result, in the next calculation. I round only the final result. When I need to write down a decimal result for later use, then I always round to two places more than the number of places I need in the final answer, to eliminate round-off error. Cheers

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[Ventus]

Thanks. Dr Otis my exam was based on both Pythagoras and Sine Cosine and Tangent. In my school we use Pythagoras and SOHCAHTOA for trigonometry. Secondly here are my answers for each question.
1. What was the distance covered by both planes in 30 minutes?
Aeroplane A- 173.21 Km
Aeroplane B- 226.58 Km
For this question I drew a diagram where the hypotenuse was 200 and 250
2. Find the altitude of both planes after 30 minutes
Aeroplane A- 99.99 Km
Aeroplane B- 105.65 Km
3. What is the difference in amplitude?
The difference is 5.66 Km
(Also each questions asks to round of to 2 decimal places.)

 

[Otis]

... Dr Otis my exam was based on both Pythagoras and Sine Cosine and Tangent. In my school we use Pythagoras ... [also] for trigonometry ...

Thank you for the added clarification. In that case, using the Pythagorean Theorem (and not sine) seems okay.

By the way, I am not a doctor.

... the distance covered by [each plane] ...
Aeroplane A- 173.21 Km
Aeroplane B- 226.58 Km

Correct.

It's not a big deal (in this exercise), but the standard abbreviation for kilometers is km. (Upper-case K represents Kelvin.)

... the altitude of [each plane] ...
Aeroplane A- 99.99 Km
Aeroplane B- 105.65 Km

The altitude of plane A is 100 km

Without seeing your new work, I'm not sure why you still have a rounding error.

Here's my method: First, memorize the exact sine and cosine values for the special angles. Then:

sin(30º) = 1/2

From your right triangle representing plane A, we get:

sin(30º) = opposite/200

opposite = 200 ∙ sin(30º)

opposite = 200 ∙ 1/2

That's 100.

... What is the difference in amplitude?
The difference is 5.66 Km ...

That ought to be 'altitude'.

After fixing the rounding error in part (2), you'll get the correct answer to part (3):

5.65 km

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[Ventus]

Yeah I know that. But the reason I got 99.99 because I did the square root of 200 squared minus 173.21 squared which equals to 99.99147914. Which rounds of 99.99. It happened since I used Pythagoras instead of sine. I hope that clears things Otis

 

[Otis]

... I hope that clears things Otis

We're getting closer to clarity, heh.

... the reason I got 99.99 [is] because I did the square root of (200 squared minus 173.21 squared) ... since I used Pythagoras instead of sine ...

Using the Pythagorean Theorem is not the issue.

The issue is that you rounded an intermediate result. Please re-read the discussion about round-off error, in posts #3 and #4.

Were I to have used the Pythagorean Theorem, then I would have used 173.2051 instead of 173.21 because (knowing that the final answer needs to be reported to two places) I would have rounded all intermediate results to four places -- to prevent round-off error.

√[200^2 - (173.2051)^2]

√[40000 - 30000.0067]

√9999.9933

Cheers

Calculator evaluation: 99.99996665

Rounding for final answer: 100

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[Ventus]

Thanks . You were helpful. I understand why I got 99.99 instead of 100. So I understand the reason I got different results was because of my roundings. I just would like to seek one more advice. I recently did my math exam and Im only worried about this question. How would I convince my teacher to give me the whole marks? I think the question may have been out of 4 or 5. I really want to make it to compression in my school which is pretty much Advanced Maths and study 2 units. How would you handle this situation if you were me? Thanks

 

[Otis]

...... How would I convince my teacher to give me the whole marks? ...

I can't advise you about that because I don't have all the information. From experience, I can say that students don't usually get full credit when they make mistakes; it depends on all the circumstances.

I wish you good fortune.

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