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3 years ago

Solve the proportion ((x-8)/5)=(2/4)

Mathematics

2 answers:

stiks02 [169]3 years ago

8 0

Cross multiply.

4(x - 8) = 5(2)

4x - 32 = 10

4x = 42

x = 42/4

x = 21/2

x = 10.5

Vika [28.1K]3 years ago

4 0

4(x - 8) = (2)(5)
4x - 32 = 10
4x = 42
x = 10.5

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so that

$\displaystyle 768 \iint_R u^2 \, du \, dv = 768 \int_{-1}^1 \int_{-\sqrt{1-v^2}}^{\sqrt{1-v^2}} u^2 \, du \, dv$

Now you could evaluate the integral as-is, but it's really much easier to do if we convert to polar coordinates.

$\begin{cases} u = r\cos(\theta) \\ v = r\sin(\theta) \\ u^2+v^2 = r^2\\ du\,dv = r\,dr\,d\theta\end{cases}$

Then

$\displaystyle 768 \int_{-1}^1 \int_{-\sqrt{1-v^2}}^{\sqrt{1-v^2}} u^2\,du\,dv = 768 \int_0^{2\pi} \int_0^1 (r\cos(\theta))^2 r\,dr\,d\theta \\\\ ~~~~~~~~~~~~ = 768 \left(\int_0^{2\pi} \cos^2(\theta)\,d\theta\right) \left(\int_0^1 r^3\,dr\right) = \boxed{192\pi}$

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2 years ago

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Answer:

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