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Gala2k [10]
3 years ago
7

Rick was given a 15% tip waiting on a table of customers who ordered $60.00 in food and drinks. How much did Rick earn in tips f

or waiting on this table?
Mathematics
1 answer:
vfiekz [6]3 years ago
4 0

Answer:

hi

Step-by-step explanation:

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<img src="https://tex.z-dn.net/?f=%20%5Crm%20%5Cint_%7B0%7D%5E%7B%20%20%5Cpi%20%7D%20%5Ccos%28%20%5Ccot%28x%29%20%20%20%20-%20%2
Nikolay [14]

Replace x with π/2 - x to get the equivalent integral

\displaystyle \int_{-\frac\pi2}^{\frac\pi2} \cos(\cot(x) - \tan(x)) \, dx

but the integrand is even, so this is really just

\displaystyle 2 \int_0^{\frac\pi2} \cos(\cot(x) - \tan(x)) \, dx

Substitute x = 1/2 arccot(u/2), which transforms the integral to

\displaystyle 2 \int_{-\infty}^\infty \frac{\cos(u)}{u^2+4} \, du

There are lots of ways to compute this. What I did was to consider the complex contour integral

\displaystyle \int_\gamma \frac{e^{iz}}{z^2+4} \, dz

where γ is a semicircle in the complex plane with its diameter joining (-R, 0) and (R, 0) on the real axis. A bound for the integral over the arc of the circle is estimated to be

\displaystyle \left|\int_{z=Re^{i0}}^{z=Re^{i\pi}} f(z) \, dz\right| \le \frac{\pi R}{|R^2-4|}

which vanishes as R goes to ∞. Then by the residue theorem, we have in the limit

\displaystyle \int_{-\infty}^\infty \frac{\cos(x)}{x^2+4} \, dx = 2\pi i {} \mathrm{Res}\left(\frac{e^{iz}}{z^2+4},z=2i\right) = \frac\pi{2e^2}

and it follows that

\displaystyle \int_0^\pi \cos(\cot(x)-\tan(x)) \, dx = \boxed{\frac\pi{e^2}}

7 0
2 years ago
What are scale drawings? can you give an example?
Alecsey [184]
Scale Drawings are drawings that are used to show the true size of something.
Scale drawings are most commonly used in maps, or in large scale drawings. These show the scale of something. It may show that 1cm is equivalent to 1km, which would allow someone to measure the map to see how far the distance it. It also allows a map to be made smaller, and less detailed- making it often easier to read.
Hope this helps :) 
7 0
3 years ago
SOMEONE PLEASE ACTUALLY HELP ME!!!!
ser-zykov [4K]

Answer:

with what do u need help with?

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