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.

personguy Jun 14, 2020

#1**+2 **

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.

thelizzybeth Jun 14, 2020

#2**0 **

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**+1 **

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

#4**0 **

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**+1 **

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**0 **

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**+1 **

Ok, so in that case, just replace 19 with 19/4 and calculate the answer. Sorry about that!

thelizzybeth
Jun 14, 2020