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iren [92.7K]
3 years ago
9

Show your work please answer asap! 40 points.

Mathematics
1 answer:
olga nikolaevna [1]3 years ago
4 0

Answer:

1. Non equivalent, use the distributive property. 5*3=18, 5*4=20, so the equation is now 18y - 20, which is not equal to 15y - 20

2. Same with this one, -4 * -2p = 8p, and  -4 * 4 = -16. Now the equation is 8p - (-16), or 8p + 16. Not equivalent.

3. 3*-7x = -21x, and 3*13 = 39. Now the equation is -21x - 39, not equivalent.

4. -6*-4a = 24a, and -6*8 = -48, the equation is now 24a+ (-48), or 24a - 48

I did my best, and I am confident in the answers, but I apologize if I am wrong.

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The slop of the line that is represented by the equation -5x + 2y = 10 is \frac{5}{2}
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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}}

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Answer:

4 feet of wall should be on each side of the painting.

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(Total wide of the wall - wide of the painting) / 2 = X

(12 - 4) / 2 = X

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4 = X

Therefore, 4 feet of wall should be on each side of the painting.

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