# Hard thinking questions

1. Oct 21, 2010

### I Like Pi

1. The problem statement, all variables and given/known data

a) given, a + 5 and 2a-1, one is 40% greater than the other one, solve for a.

b) given, 5x$$^{4}$$ + 4x$$^{3}$$ + 3x$$^{2}$$ + Px + Q. When divided by x$$^{2}$$ - 1, you get a remainder of 0, solve for p and q.

c) given csc(6b + $$\frac{pi}{8}$$) = sec(2b - $$\frac{pi}{8}$$), solve for b.

2. Relevant equations

3. The attempt at a solution

please help, i can't get started on these three at all...

edit:

c) i know that cscx = 1/sinx and secx = 1/cosx, i then cross multiplied to get $$cos(2b-pi/8) = sin(6b+pi/8)... Last edited: Oct 21, 2010 2. Oct 21, 2010 ### Quinzio Question a) shouldn't be a problem at all. Any attempt by you ? 3. Oct 21, 2010 ### I Like Pi Would it be (a+5) = .4(2a-1) + 2a-1 ? or .4(a+5) + (a+5) = 2a - 1 ? 4. Oct 21, 2010 ### Quinzio It's not clear from the text. Choose one and go on, or solve both in sequence. 5. Oct 21, 2010 ### I Like Pi well then you get a+5 = 0.8a - 0.4 + 2a -1 a+5 = 2.8a - 1.4 1.8a - 6.4 = 0 1.8a = 6.4 a = 3.55555556 or 2a - 1 = .4a + 2 + a + 5 2a - 1 = 1.4a + 7 0.6a - 8 = 0 0.6a = 8 a = 13.3333333 therefore a can be 13.333 or 3.555 ? 6. Oct 21, 2010 ### Quinzio yes. 7. Oct 21, 2010 ### I Like Pi thanks could you help me get started on b) ? Thanks for your time! 8. Oct 21, 2010 ### Quinzio b) Make this multiplication [tex] (x^2-1)(Ax^2+Bx+C)$$

9. Oct 21, 2010

### I Like Pi

that would give you $$ax^4 + bx^3 + (c - a)x^2 - bx - c$$. Is that what you mean?

10. Oct 21, 2010

### Quinzio

Ok, so now give the solution. It's really easy.

11. Oct 21, 2010

### I Like Pi

do you use the original equation? so a = 5, b = 4, and c = 3?

but that doesn't work... cause then you get:$$5x^4 + 4x^3 + (3 - 5)x^2 - 4x - 3$$

12. Oct 21, 2010

### Quinzio

You realize by yourself there's something wrong. Ok.

You should come alone to the solution. The next step would be giving you the solution.

13. Oct 21, 2010

### I Like Pi

I don't see what I could possibly do to get the answer I know that c - a = 3, that's all... but c itself is 3

14. Oct 21, 2010

### Quinzio

Why you say "c itself is 3"... ???
Tell me where is written that ?

15. Oct 21, 2010

### I Like Pi

oh, well i based it on the original... a = 5, b = 4, c = 3...

well, then if that's the case,

$$ax^4 + bx^3 + (c - a)x^2 - bx - c$$
$$=5x^4 + 4x^3 + (c - 5)x^2 - 4x - c$$
therefore
$$c = 8$$?
and therefore:
$$P = -4$$
$$Q = -8$$

is that what you meant?

Thanks for your help! Means a lot!

Last edited: Oct 21, 2010
16. Oct 21, 2010

### zgozvrm

Not quite....
You solve for the cases where one term is 40% of the other, not 40% greater then the other, as the question asked!

In other words, suppose you have a number (let's say 20) and you want to find the value that is 40% greater.
0.4 * 20 = 8 which is clearly not 40% greater. In fact, it is less!

What would you do to determine the value that is 40% greater than 20?

17. Oct 21, 2010

### I Like Pi

Well you would do .4 * 20 + 20, a value 40% greater then 20 is 28?

18. Oct 21, 2010

### zgozvrm

Yes, but that simplifies...

0.4 * 20 + 20 = (0.4 + 1) * 20 = 1.4 * 20

Therefore, a number that is 40% larger than x is 1.4 * x

19. Oct 21, 2010

### 94JZA80

well at first glance (and i do mean i only "glanced" at it for a split second), i thought the same thing...here are the equations he set up:
look closer at the right-hand side of the first equation. 0.4(2a-1) + 2a-1 = 1.4(2a-1), and the original equation becomes a+5 = 1.4(2a-1), which certainly implies that a+5 is 40% larger than 2a-1. likewise, we can see that the left-hand side of the 2nd equation, 0.4(a+5) + (a+5), equals 1.4(a+5). and so equation 2 becomes 1.4(a+5) = 2a-1, which certainly implies that 2a-1 is 40% greater than a+5.

your calculations are correct - you just chose to represent the quantity on one side ofthe equation as 0.4x + x instead of 1.4x.

20. Oct 21, 2010

### tiny-tim

Hi I Like Pi!!

(since you like them so much, have a pi: π )
Yes, and now use cosθ = sin(π/2 - θ) to get that in the form sinA = sinB.

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