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2 years ago

Write as a fraction 37.5%

1 answer:

8 0

If you check Mathway it shows you how to convert 37.5% into a fraction, 3/8, but "<span>The number 37.5 is expressed as 37 1/2 in simplified terms or 75/2 as a fraction." is the most common answer shown on the web.</span>

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If the range of the function f(x)=x/4 is {28, 30, 32, 34, 36}, what is its domain?

avanturin [10]

Y = x/4
x = 4y
domain = {4(28), 4(30), 4(32), 4(34), 4(36)} = {112, 120, 128, 136, 144}

4 0

2 years ago

<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

1 year ago

One-half the square of b

enot [183]

A machine produces 240 bolts in 25 minutes. At the same rate how many bolts would be produced in 45 minutes

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2 years ago

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After Ricardo received his allowance for the week, he went to the mall with some friends. He spent half of his allowance on a ne

ad-work [718]

Lets work backwards, he had $5 after it all, and spent $1.25 on a snack, so we add that to the remainder, which is $6.25. then he spent half of that on whatever stuff he likes, so add $6.25 and $6.25, which is $12.50

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2 years ago

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Jessica gets her favorite shade of purple paint by mixing 2 cups of blue paint and 3 cups of red paint

Pie

Ok but what’s the question

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2 years ago

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