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diamong [38]
3 years ago
13

What’s the answer to this

Mathematics
1 answer:
Aliun [14]3 years ago
7 0

Answer:

v=9

Step-by-step explanation:

50 + x =180

x=130

130+(6v-4)= 180 (subtract 126 from each side)

6v=54 (divide by 6)

v=9

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Question (c)! How do I know that t^5-10t^3+5t=0?<br> Thanks!
astra-53 [7]
(a) By DeMoivre's theorem, we have

(\cos\theta+i\sin\theta)^5=\cos5\theta+i\sin5\theta

On the LHS, expanding yields

\cos^5\theta+5i\cos^4\theta\sin\theta-10\cos^3\theta\sin^2\theta-10i\cos^2\theta\sin^3\theta+5\cos\theta\sin^4\theta+i\sin^4\theta

Matching up real and imaginary parts, we have for (i) and (ii),


\cos5\theta=\cos^5\theta-10\cos^3\theta\sin^2\theta+5\cos\theta\sin^4\theta
\sin5\theta=5\cos^4\theta\sin\theta-10\cos^2\theta\sin^3\theta+\sin^5\theta

(b) By the definition of the tangent function,

\tan5\theta=\dfrac{\sin5\theta}{\cos5\theta}
=\dfrac{5\cos^4\theta\sin\theta-10\cos^2\theta\sin^3\theta+\sin^5\theta}{\cos^5\theta-10\cos^3\theta\sin^2\theta+5\cos\theta\sin^4\theta}

=\dfrac{5\tan\theta-10\tan^3\theta+\tan^5\theta}{1-10\tan^2\theta+5\tan^4\theta}
=\dfrac{t^5-10t^3+5t}{5t^4-10t^2+1}


(c) Setting \theta=\dfrac\pi5, we have t=\tan\dfrac\pi5 and \tan5\left(\dfrac\pi5\right)=\tan\pi=0. So

0=\dfrac{t^5-10t^3+5t}{5t^4-10t^2+1}

At the given value of t, the denominator is a non-zero number, so only the numerator can contribute to this reducing to 0.


0=t^5-10t^3+5t\implies0=t^4-10t^2+5

Remember, this is saying that

0=\tan^4\dfrac\pi5-10\tan^2\dfrac\pi5+5

If we replace \tan^2\dfrac\pi5 with a variable x, then the above means \tan^2\dfrac\pi5 is a root to the quadratic equation,

x^2-10x+5=0

Also, if \theta=\dfrac{2\pi}5, then t=\tan\dfrac{2\pi}5 and \tan5\left(\dfrac{2\pi}5\right)=\tan2\pi=0. So by a similar argument as above, we deduce that \tan^2\dfrac{2\pi}5 is also a root to the quadratic equation above.

(d) We know both roots to the quadratic above. The fundamental theorem of algebra lets us write

x^2-10x+5=\left(x-\tan^2\dfrac\pi5\right)\left(x-\tan^2\dfrac{2\pi}5\right)

Expand the RHS and match up terms of the same power. In particular, the constant terms satisfy

5=\tan^2\dfrac\pi5\tan^2\dfrac{2\pi}5\implies\tan\dfrac\pi5\tan\dfrac{2\pi}5=\pm\sqrt5

But \tanx>0 for all 0, as is the case for x=\dfrac\pi5 and x=\dfrac{2\pi}5, so we choose the positive root.
3 0
3 years ago
Give an example of a quadratic with complex roots. Explain how you know it has complex roots.
Vladimir [108]
2x^2 + x + 3=0 has only complex roots.

The determinant is 1-4*2*3 = -23

-23 (or any determinant) is the part under the square root sign,  If that determinant is negative, knowing you cannot take the square root of a negative number, we know the answers must be complex.
5 0
3 years ago
The equation of the graph line is 2X-3Y=12
Klio2033 [76]

Answer:

A. -4

Step-by-step explanation:

For solving for x intercepts analytically. You can set the the y in the equation to 0. So, 2x-3(0)=12, and solving for x will get you -4.

You can also solve graphically by plugging in the equation and looking at where it intercepts the x axis.

3 0
3 years ago
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MatroZZZ [7]
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5 0
3 years ago
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Last Wednesday, a random sample of 24 students were surveyed to find how long it takes to walk from the Fretwell Building to the
Ray Of Light [21]

Answer:

E

Step-by-step explanation:

Solution:-

- We are to investigate the confidence interval of 95% for the population mean of walking times from Fretwell Building to the college of education building.

- The survey team took a sample of size n = 24 students and obtained the following results:

                Sample mean ( x^ ) = 12.3 mins

                Sample standard deviation ( s ) = 3.2 mins

- The sample taken was random and independent. We can assume normality of the sample.

- First we compute the critical value for the statistics.

- The z-distribution is a function of two inputs as follows:

  • Significance Level  ( α / 2 ) = ( 1 - CI ) / 2 = 0.05/2 = 0.025

Compute: z-critical = z_0.025 = +/- 1.96

- The confidence interval for the population mean ( u ) of  walking times is given below:

                      [ x^ - z-critical*s / √n  ,   x^ + z-critical*s / √n  ]

Answer:        [ 12.3 - 1.96*3.2 / √24  ,  12.3 + 1.96*3.2 / √24  ]

                   

3 0
3 years ago
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