Where R is the median between Q and L:

From my understanding of a triangle's centroid, it divides an angle bisector into parts of 2/3 and 1/3. In the given problem, these divisions are NS and SR. Therefore, twice SR would be equal to NS. From here, we can get the value of X, to solve for SR.

NS = 2SR
(x + 10) = 2(x + 3)
x + 10 = 2x + 6
x = 4

Therefore, SR = (x + 3) = 7

4 0

2 years ago

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Use Green's Theorem to calculate the circulation of F =2xyi around the rectangle 0≤x≤8, 0≤y≤3, oriented counterclockwise.

Tamiku [17]

Green's theorem says the circulation of $\vec F$ along the rectangle's border $C$ is equal to the integral of the curl of $\vec F$ over the rectangle's interior $D$ .

Given $\vec F(x,y)=2xy\,\vec\imath$ , its curl is the determinant

$\det\begin{bmatrix}\frac\partial{\partial x}&\frac\partial{\partial y}\\2xy&0\end{bmatrix}=\dfrac{\partial(0)}{\partial x}-\dfrac{\partial(2xy)}{\partial y}=-2x$

So we have

$\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\iint_D-2x\,\mathrm dx\,\mathrm dy=-2\int_0^3\int_0^8x\,\mathrm dx\,\mathrm dy=\boxed{-192}$

6 0

2 years ago