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Chapter 10: Problem 12

Given the cylinder defined by \(x^{2}+y^{2}=4\), find the path of shortest length connecting the given points. a. \((2,0,0)\) and \((0,2,5)\) b. \((2,0,0)\) and \((2,0,5)\)

Short Answer

Expert verified
The shortest path for path a is \(\sqrt{(π)^2 + 25}\) and for path b is 5.

Step by step solution

01

Path a

First, solve for the path a from \((2,0,0)\) to \((0,2,5)\). We need to solve for the shortest distance between these two points in terms of the angular component \(θ\) and the height \(z\). To get from point a to b we need to go an angle of \(π/2\) around the cylinder and up 5 units. Thus, the shortest path for a is \(d = \sqrt{((π/2)*2)^2 + 5^2} = \sqrt{(π)^2 + 25}\)
02

Path b

Next, solve for the path b from \((2,0,0)\) to \((2,0,5)\). Here, there is no change in the angular component \(θ\). The shortest path is thus only due to a change in the height \(z\), giving a shortest path \(d = 5\).
03

Final minimization conclusion

Thus the shortest path for each point in the cylinder is a distance of \(\sqrt{(π)^2 + 25}\) for path a and \(5\) for path b.

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Most popular questions from this chapter

Find the extrema of the given function subject to the given constraint. a. \(f(x, y)=(x+y)^{2}, x^{2}+y=1\) b. \(f(x, y)=x^{2} y+x y^{2}, x^{2}+y^{2}=2\) c. \(f(x, y)=2 x+3 y, 3 x^{2}+2 y^{2}=3\) d. \(f(x, y, z)=x^{2}+y^{2}+z^{2}, x y z=1\) e. \(f(x, y, z)=x y+y x, x^{2}+y^{2}=1, x z=1\)

For each problem, locate the critical points and classify each one using the second derivative test. a. \(f(x, y)=(x+y)^{2}\). b. \(f(x, y)=x^{2} y+x y^{2}\). c. \(f(x, y)=x^{4} y+x y^{4}-x y .\) d. \(f(x, y)=x^{2}-3 x y+2 x+10 y+6 y^{2}+12\). e. \(f(x, y)=\left(x^{2}-y^{2}\right) e^{-y}\)

Find the absolute maxima and minima of the function \(f(x, y)=x^{2}+\) \(x y+y^{2}\) on the unit circle.

A light ray travels from point A in a medium with index of refraction \(n_{1}\) toward point \(\mathrm{B}\) in a medium with index of refraction \(n_{2}\). Assume that the a. Write the time functional in terms of the travel path \(y(x)\). b. Apply Fermat's Principle of least time to write the Euler Equation for this functional. c. Solve the equation in part b. and show that \(n \sin \theta\) is a constant. d. Let the point at which the light is incident to the interface be at \((x, 0)\). Write an expression for the total time to travel from point \(A\) at \(\left(x_{1}, y_{1}\right)\) to point \(B\) at \(\left(x_{2}, y_{2}\right)\) in terms of the indices of refraction. and the coordinates \(x, x_{1}, x_{2}\), and \(y_{2}\). e. Treating the time as a function of \(x\), minimize this function as a function of one variable and derive Snell's law of refraction.

For each of the following, find a path that extremizes the given integral. a. \(\int_{1}^{2}\left(y^{\prime 2}+2 y y^{\prime}+y^{2}\right) d y, y(1)=0, y(2)=1\) b. \(\int_{0}^{3} y^{2}\left(1-y^{2}\right)^{2} d y, y(0)=1, y(2)=2\) c. \(\int_{-1}^{1} 5 y^{\prime 2}+2 y y^{\prime} d y, y(-1)=1, y(1)=0\)
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