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Over [174]
3 years ago
14

Consider the force field and circle defined below.

Mathematics
1 answer:
grin007 [14]3 years ago
4 0

By Green's theorem,

\displaystyle\int_{x^2+y^2=9}\vec F(x,y)\cdot\mathrm d\vec r=\iint_D\left(\frac{\partial(xy)}{\partial x}-\frac{\partial(x^2)}{\partial y}\right)\,\mathrm dx\,\mathrm dy=\iint_Dy\,\mathrm dx\,\mathrm dy

where C is the circle x^2+y^2=9 and D is the interior of C, or the disk x^2+y^2\le1.

Convert to polar coordinates, taking

\begin{cases}x=r\cos\theta\\y=r\sin\theta\end{cases}\implies\mathrm dx\,\mathrm dy=r\,\mathrm dr\,\mathrm d\theta

Then the work done by \vec F on the particle is

\displaystyle\iint_Dy\,\mathrm dx\,\mathrm dy=\int_0^{2\pi}\int_0^3(r\sin\theta)r\,\mathrm dr\,\mathrm d\theta=\left(\int_0^{2\pi}\sin\theta\,\mathrm d\theta\right)\left(\int_0^3r^2\,\mathrm dr\right)=\boxed0

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The value of k is (A) 20/3.

<h3>What is a parabola?</h3>
  • A parabola is a planar curve that is mirror-symmetrical and roughly U-shaped in mathematics.
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To find the value of k:

  • y - b = c (x - a)²

Where (a, b) is the vertex and c is the constant.

  • (a, b) = (-4, k)
  • y - k = c (x - (- 4))²
  • y - k = c (x + 4)²

So,

  • x = 0, y = 12
  • 12 - k = 16c
  • k = 12 - 16c  ...... (1)

Then,

  • (-3, 7) = (x, y)
  • 7 - k = c (1)²
  • k = 7 - c  ...... (2)

Now,

  • 12 - 16c = 7 - c
  • 12 - 7 = 16c - c
  • 5 = 15c
  • c = 5/15 = 1/3

So, the value of k:

  • k = 12 - 16 (1/3) = 12 - 16/3 = 36-16/3 = 20/3

Therefore, the value of k is (A) 20/3.

Know more about a parabola here:

brainly.com/question/4061870

#SPJ4

The complete question is given below:

The coordinates of the vertex of a parabola in the XY plane are (-4,k). If the y-intercept of the parabola is 12 and the parabola passes through the point (-3,7), then what is the value of k?

(A) 20/3

(B) 16/5

(C) 14/3

(D) 12/5

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