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3 years ago

Express answer in exact form.

Mathematics

2 answers:

zepelin [54]3 years ago

7 0

I'm guessing that you want to find the segment area of a circle that has a radius AO = 8" and a chord AB with a length of 8".

Sine angle AOD = AE / OA
Sine angle AOD = 4 / 8
Sine angle AOD = .5
arc sine (.5) = 30 degrees
So, angle AOB = 60 degrees

Circle Area = PI * radius^2
Circle Area = 201.06
Sector Area = (60/360) * 201.06
Sector Area = 33.51

Line OE^2 = AO^2 -AE^2
Line OE^2 = 64 -16
Line OE = 6.9282032303 

Triangle AOB Area = OE*AE = 6.9282032303 * 4
Triangle AOB Area = 27.7128129211 

Segment Area = Sector Area -Triangle AOB Area
Segment Area = 33.51 -27.71
Segment Area = 5.80

Sedaia [141]3 years ago

6 0

A=(64/6 pie 18~3) inches 2

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Find all the complex roots. Write the answer in exponential form.

dezoksy [38]

We have to calculate the fourth roots of this complex number:

$z=9+9\sqrt[]{3}i$

We start by writing this number in exponential form:

$\begin{gathered} r=\sqrt[]{9^2+(9\sqrt[]{3})^2} \\ r=\sqrt[]{81+81\cdot3} \\ r=\sqrt[]{81+243} \\ r=\sqrt[]{324} \\ r=18 \end{gathered}$ $\theta=\arctan (\frac{9\sqrt[]{3}}{9})=\arctan (\sqrt[]{3})=\frac{\pi}{3}$

Then, the exponential form is:

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The formula for the roots of a complex number can be written (in polar form) as:

$z^{\frac{1}{n}}=r^{\frac{1}{n}}\cdot\lbrack\cos (\frac{\theta+2\pi k}{n})+i\cdot\sin (\frac{\theta+2\pi k}{n})\rbrack\text{ for }k=0,1,\ldots,n-1$

Then, for a fourth root, we will have n = 4 and k = 0, 1, 2 and 3.

To simplify the calculations, we start by calculating the fourth root of r:

$r^{\frac{1}{4}}=18^{\frac{1}{4}}=\sqrt[4]{18}$

NOTE: It can not be simplified anymore, so we will leave it like this.

Then, we calculate the arguments of the trigonometric functions:

$\frac{\theta+2\pi k}{n}=\frac{\frac{\pi}{2}+2\pi k}{4}=\frac{\pi}{8}+\frac{\pi}{2}k=\pi(\frac{1}{8}+\frac{k}{2})$

We can now calculate for each value of k:

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Answer:

The four roots in exponential form are

z0 = 18^(1/4)*e^(i*π/8)

z1 = 18^(1/4)*e^(i*5π/8)

z2 = 18^(1/4)*e^(i*9π/8)

z3 = 18^(1/4)*e^(i*13π/8)

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