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Alex777 [14]
3 years ago
12

Use stokes' theorem to evaluate c f · dr where c is oriented counterclockwise as viewed from above. f(x, y, z = xyi + 5zj + 7yk,

c is the curve of intersection of the plane x + z = 5 and the cylinder x2 + y2 = 36.
Mathematics
1 answer:
Helga [31]3 years ago
4 0
The intersection can be parameterized by

C:=\mathbf r(t)=\begin{cases}x(t)=6\cos t\\y(t)=6\sin t\\z(t)=5-6\cos t\end{cases}

with 0\le t.

By Stoke's theorem, the integral of \mathbf f(x,y,z)=xy\,\mathbf i+5z\,\mathbf j+7y\,\mathbf k along C is equivalent to

\displaystyle\int_C\mathbf f(x(t),y(t),z(t))\cdot\mathrm d\mathbf r(t)=\iint_S\nabla\times\mathbf f\,\mathrm dS

where S is the region bounded by C. The line integral reduces to

\displaystyle\int_0^{2\pi}(36\sin t\cos t\,\mathbf i+(25-30\cos t)\,\mathbf j+42\sin t\,\mathbf k)\cdot(-6\sin t\,\mathbf i+6\cos t\,\mathbf j+6\sin t\,\mathbf k)\,\mathrm dt
=\displaystyle\int_0^{2\pi}(54(\cos3t-\cos t)-30(3\cos2t-5\cos t+3)+(126-126\cos2t)\,\mathrm dt
=\displaystyle\int_0^{2\pi}(36+96\cos t-216\cos2t+54\cos3t)\,\mathrm dt
=72\pi
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volume \ of \ pipe = \pi r^2l\\volume \ of \ pipe = \3.14* 2^2*17 \\volume \ of \ pipe =  213.52

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Answer: You need to wait at least 6.4 hours to eat the ribs.

t ≥ 6.4 hours.

Step-by-step explanation:

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after the first hour, at h = 1h we have an increase of 25%, this means that the new temperature is:

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after another hour we have another increase of 25%, the temperature now is:

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