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I have finished the rest of the problems from my last post myself.

 

Here are the rest, I am stuck on.

 

1. In triangle ABC, sin A : sin B : sin C = 2 : 3 : 4. Find cos(A+C).

 

2. The combined area of all the faces of a rectangular prism is 12, and the combined length of all its edges is 19. Find the length of a space diagonal of the prism. 

 Jun 14, 2020
 #1
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I'm not too sure how to do number 1 (not that great with trig) but here's number 2. 

 

2. Let's write out the information we have so far. Set a, b, and c as the three sides.

a+b+c=19/4

2(ab+ac+bc)=12

 

We are trying to figure out the space diagonal. We see that a space diagonal equals \(\sqrt{a^2+b^2+c^2}\) through the Pythagorean Theorem (to see why, draw a diagram. There is a right triangle between the height of the prism, the base diagonal, and the space diagonal.)

 

Now let's square both sides of a+b+c=19/4, which after expanding, simplifies to \(a^2+b^2+c^2+2ab+2ac+2bc=\frac {361} {16}\)

Substituting in 2(ab+ac+bc)=12, the second equation at the top, the equation becomes \(a^2+b^2+c^2+12=\frac{361} {16}\). Simplifying, we find\(a^2+b^2+c^2=\frac{169}{16}\) 

We are trying to find the square root of this, so our answer is \(\frac{13}{2}\)

 

EDIT: I have corrected my solution.

 Jun 14, 2020
edited by thelizzybeth  Jun 14, 2020
edited by thelizzybeth  Jun 17, 2020
 #2
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Thank you, but I'm a little confused...
Which sides are a, b, and c? Why are there only 3 sides? Isn't it a rectangular prism?

personguy  Jun 14, 2020
 #3
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Yes, there are twelve edges, but four are the same length, so we can express those lengths as a, b, and c. 

Just like for a rectangle, there are four edges, but we only need two variables: L and W to express the length and width. 

Now that you asked this question though, I am wondering if "the combined length of all its edges is 19." means 4(a+b+c)=19. I assumed it meant different edges (why I only did a+b+c), but I may be wrong there. 

thelizzybeth  Jun 14, 2020
edited by thelizzybeth  Jun 14, 2020
edited by thelizzybeth  Jun 14, 2020
 #4
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Oh well it turns out that "all of the edges" is every edge not every different one at least I think.

personguy  Jun 14, 2020
 #5
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Where is this question from? Is it from somewhere online like AOPS' Alcumus where you can see if you got it wrong? Or is it like homework?

thelizzybeth  Jun 14, 2020
 #6
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It's from AOPS, but it never says the combined length of all different edges, it says the combined length of all its edges.

personguy  Jun 14, 2020
 #7
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Ok, so in that case, just replace 19 with 19/4 and calculate the answer. Sorry about that!

thelizzybeth  Jun 14, 2020
 #8
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No problem, thank for helping!

personguy  Jun 14, 2020
 #10
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I edited my answer.

thelizzybeth  Jun 14, 2020
 #9
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1. In triangle ABC, sin A : sin B : sin C = 2 : 3 : 4. Find cos(A+C).

 

Angle A = 29º          sin(A) ≈ 0.485

 

Angle B = 46.653º     sin(B) ≈ 0.727

 

Angle c = 104.347º     sin(C) ≈ 0.969

 

cos(A+C) = -0.6864     smiley

 Jun 14, 2020

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