Please help please help

I could not think of an easy way to solve this, apart from having a graphing calculator do it. In the end, I found I could solve it analytically using a combination of the law of sines and the law of cosines.

Let x represent the length of the shortest side, and θ the smallest angle. Then the law of sines tells you ...

sin(θ)/x = sin(2θ)/(x+2)

Cross-multiplying and using the trig identity for sin(2θ), we have ...

(x +2)sin(θ) = 2x·sin(θ)cos(θ)

Dividing out sin(θ), we see that ...

cos(θ) = (x+2)/(2x)

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The law of cosines for the shortest side and smallest angle tells you ...

x^2 = (x+1)^2 + (x+2)^2 - 2(x+1)(x+2)·cos(θ)

Substituting the above expression for cos(θ), this can be rewritten as ...

0 = (x^2 +2x +1) +(x^2 +4x +4) -x^2 -(x+1)(x+2)^2/x

0 = x^2 +6x +5 -(x+1)(x+2)^2/x . . . . . . collect terms outside the fraction

0 = x(x+5)(x+1) -(x+1)(x+2)^2 . . . . . . . . factor and multiply by x

We know that x=-1 is not a solution, so we can divide by that factor:

0 = x^2 +5x -(x^2 +4x +4) . . . . . multiply it all out

0 = x -4 . . . . . . . . . . . . . . . . . . . . collect terms

4 = x

so, cos(θ) = (4+2)/(2·4) = 6/8 = 3/4

and the angle of interest is ...

θ = arccos(3/4) ≈ 41.40962° ≈ 41°

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The attachment shows a triangle-solver's result using the consecutive integers for side lengths. It confirms the answer we have here.

7 0

3 years ago

Solve the proportion. 8/7 = n/9

RoseWind [281]

$\frac{8}{7} = \frac{n}{9}$

Cross multiply the fractions:

$7 \times n = 7n$
$8 \times 9 = 72$

$7n = 72$

Divide both sides by 7 to get n by itself:

$72 \div 7 = 10.28$
$n = 10.28$

The value of n is 10.28.

5 0

3 years ago

Factor the following problems SHOW YOUR WORK 1. x^2+16x+24 2. x^2-10a+25

nalin [4]

x^2+16x+24: Think: What are the integer factors of 24? I'd list 1, 2, 3, 4, 6, 8, 12 and 24. Next: Look for a pair of factors from that list that:

(1)ADD up to 16 AND (2) whose product is 24. Unfortunately there seems to be no such pair from the above list.

So: Use the quadratic formula to find the roots of x^2+16x+24:

-16 plus or minus sqrt[ 256-4(1)(24) ]
x = --------------------------------------------------
2
-16 plus or minus sqrt(160) -16 plus or minus 4 sqrt(10)
= --------------------------------------- = ----------------------------------------
2 2
2(-8 plus or minus 2 sqrt(10) )
= ------------------------------------------- = -8 plus or minus 2sqrt(10)
2

If c is a root, (x-c) is a factor. Thus, if -8 plus 2sqrt(10) is a root,

(x+8-2sqrt(10) is a factor. Can you find the other factor?

x^2-10a+25 is much easier to factor. The sqrt of 25 is 5, so try x= -5 as a root. Dividing x-5 into x^2-10a+25, we get x-5. Therefore, the factors are

(x-5) and (x-5).

8 0

3 years ago