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Traffic engineers use road tubes to measure the number and speed of cars passing a line in the road. Suppose traffic engineers set road tubes on a freeway whose inner edge lies along the circle $x^2+y^2=0.2$and whose outer edge is $x^2+y^2=0.2113$. The traffic is flowing counterclockwise in the first quadrant of the circles. The road tubes lie along the lines $y=0$and $x=0$, and the measurements show that the speed of cars along the inner circle is 69 mph while along the outside circle it is 61 mph. Denote the stretch of road in the first quadrant by S with oriented boundary C, and denote the velocity of the cars at any point by $v(x,y)$.

(a) Compute ${鈭�}_C鈥�\mathbf{v}(x,y)鈰�d\mathbf{r}\mathbf{.}$What is the significance of the sign of this quantity?

(b) Use Green鈥檚 Theorem to compute the average curl of the traffic on this stretch of road:

$\frac{1}{A}{鈭�}_S鈥�\left(\frac{\mathrm{鈭�}{v}_2}{\mathrm{鈭�}x}鈭�\frac{\mathrm{鈭�}{v}_1}{\mathrm{鈭�}y}\right)dxdy.$

Here, A is the area of the roadway.

Step by step solution

01

Step 1. Given Information

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