+17 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.
+310 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.
+17 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?
+310 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.
+17 Oh well it turns out that "all of the edges" is every edge not every different one at least I think.
+310 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?
+17 It's from AOPS, but it never says the combined length of all different edges, it says the combined length of all its edges.
+310 Ok, so in that case, just replace 19 with 19/4 and calculate the answer. Sorry about that!
