we know that in 16.5 laps we get 37 miles, hmmm well the circumference of that circle is just 1 lap, how many miles is there in 1 lap?
![16.\underline{5}\implies \cfrac{165}{1\underline{0}}\implies \cfrac{33}{2} \\\\[-0.35em] ~\dotfill\\\\ \begin{array}{ccll} laps&miles\\ \cline{1-2} \frac{33}{2}&37\\[1em] 1&x \end{array}\implies \cfrac{~~\frac{33}{2} ~~}{1}=\cfrac{37}{x}\implies \cfrac{33}{2}=\cfrac{37}{x} \\\\\\ 33x=74\implies x=\cfrac{74}{33}\impliedby \stackrel{\textit{one lap}~\hfill }{\textit{circumference of the circle}} \\\\[-0.35em] ~\dotfill](https://tex.z-dn.net/?f=16.%5Cunderline%7B5%7D%5Cimplies%20%5Ccfrac%7B165%7D%7B1%5Cunderline%7B0%7D%7D%5Cimplies%20%5Ccfrac%7B33%7D%7B2%7D%20%5C%5C%5C%5C%5B-0.35em%5D%20~%5Cdotfill%5C%5C%5C%5C%20%5Cbegin%7Barray%7D%7Bccll%7D%20laps%26miles%5C%5C%20%5Ccline%7B1-2%7D%20%5Cfrac%7B33%7D%7B2%7D%2637%5C%5C%5B1em%5D%201%26x%20%5Cend%7Barray%7D%5Cimplies%20%5Ccfrac%7B~~%5Cfrac%7B33%7D%7B2%7D%20~~%7D%7B1%7D%3D%5Ccfrac%7B37%7D%7Bx%7D%5Cimplies%20%5Ccfrac%7B33%7D%7B2%7D%3D%5Ccfrac%7B37%7D%7Bx%7D%20%5C%5C%5C%5C%5C%5C%2033x%3D74%5Cimplies%20x%3D%5Ccfrac%7B74%7D%7B33%7D%5Cimpliedby%20%5Cstackrel%7B%5Ctextit%7Bone%20lap%7D~%5Chfill%20%7D%7B%5Ctextit%7Bcircumference%20of%20the%20circle%7D%7D%20%5C%5C%5C%5C%5B-0.35em%5D%20~%5Cdotfill)
![\textit{Circumference of a circle}\\\\ C=2\pi r~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ C=\frac{74}{33} \end{cases}\implies \cfrac{74}{33}=2\pi r\implies \cfrac{74}{33(2\pi )}=r \\\\\\ \stackrel{miles}{0.35689}\approx r\implies \stackrel{\textit{about 1884 ft and 5 inches}}{1884.395\approx r}](https://tex.z-dn.net/?f=%5Ctextit%7BCircumference%20of%20a%20circle%7D%5C%5C%5C%5C%20C%3D2%5Cpi%20r~~%20%5Cbegin%7Bcases%7D%20r%3Dradius%5C%5C%5B-0.5em%5D%20%5Chrulefill%5C%5C%20C%3D%5Cfrac%7B74%7D%7B33%7D%20%5Cend%7Bcases%7D%5Cimplies%20%5Ccfrac%7B74%7D%7B33%7D%3D2%5Cpi%20r%5Cimplies%20%5Ccfrac%7B74%7D%7B33%282%5Cpi%20%29%7D%3Dr%20%5C%5C%5C%5C%5C%5C%20%5Cstackrel%7Bmiles%7D%7B0.35689%7D%5Capprox%20r%5Cimplies%20%5Cstackrel%7B%5Ctextit%7Babout%201884%20ft%20and%205%20inches%7D%7D%7B1884.395%5Capprox%20r%7D)
Im not sure why it brought me here it just brought me here sorry
The answer is x = 51 okay bro
Since the problem is asking us to do so, we are going to use the complex conjugate formula to find the absolute value of our complex number.
The complex conjugate formula is: ![|z|=\sqrt{z(conjugate.of .z)}](https://tex.z-dn.net/?f=%20%7Cz%7C%3D%5Csqrt%7Bz%28conjugate.of%20.z%29%7D%20%20)
where
is the complex number
is the absolute value of the complex number
We know from our problem that our complex number is
, so
. Now to conjugate of our complex number, we just need to change the sign of the imaginary part:
![conjugate=8-2i](https://tex.z-dn.net/?f=%20conjugate%3D8-2i%20)
Now that we have all we need, let's replace the values in our formula:
![|z|=\sqrt{z(conjugate.of .z)}](https://tex.z-dn.net/?f=%20%7Cz%7C%3D%5Csqrt%7Bz%28conjugate.of%20.z%29%7D%20%20)
![|8+2i|=\sqrt{(8+2i)(8-2i)}](https://tex.z-dn.net/?f=%20%7C8%2B2i%7C%3D%5Csqrt%7B%288%2B2i%29%288-2i%29%7D%20%20)
![|8+2i|=\sqrt{64-16i+16i-4i^2}](https://tex.z-dn.net/?f=%20%20%7C8%2B2i%7C%3D%5Csqrt%7B64-16i%2B16i-4i%5E2%7D%20%20)
![|8+2i|=\sqrt{64-4i^2}](https://tex.z-dn.net/?f=%20%7C8%2B2i%7C%3D%5Csqrt%7B64-4i%5E2%7D%20)
![|8+2i|=\sqrt{64+4}](https://tex.z-dn.net/?f=%20%7C8%2B2i%7C%3D%5Csqrt%7B64%2B4%7D%20)
![|8+2i|=\sqrt{68}](https://tex.z-dn.net/?f=%20%7C8%2B2i%7C%3D%5Csqrt%7B68%7D%20)
![|8+2i|=2\sqrt{17}](https://tex.z-dn.net/?f=%20%7C8%2B2i%7C%3D2%5Csqrt%7B17%7D%20)
We can conclude that the absolute value of
is ![2\sqrt{17}](https://tex.z-dn.net/?f=%202%5Csqrt%7B17%7D%20%20)
Can you reword the question. I don't understand the first sentence.