You want to design a solar collector that will concentrate the sun's rays at a point....

 

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You want to design a solar collector that will concentrate the sun's rays at a point. By symmetry this surface will have a shape that is a surface of revolution obtained by revolving a curve about an axis. Let's call this the x-axis and the rays parallel to this axis are focused at the origin. We will use differential equations to derive the equation for the curve. A cross section of the solar collector showing the curve is pictured: 1. The law of reflection says that the angles y and 8 are equal. Use this and geometry to show that 3 = 20. 2. The slope of the tangent line to the curve at point (r.y) is given by = tano. We can see from the diagram that y/z= tan 3. Show that 2dy/dz 1-(dy/dr) Useful trig identities are sin(28) = 2 sin cos 8 and cos(28) = cos 6-sin²8. 3. Solve for dy/dr to find a differential equation whose solution is the curve defining the cross section of the solar collector. 4. Show that the differential equation you found is a homogeneous equation (Section 2.6). Use an appropriate variable substitution to find a separable equation. 5. Use technology to find a general solution to the differential equation and graph a few curves from the one parameter family of solutions. Attach your work (with a staple and your name!) to this worksheet. 6. Extra Credit: Use technology (or great artistic skills) to draw a 3D representation of the solar collector surface. You want to design a solar collector that will concentrate the sun's rays at a point. By symmetry this surface will have a shape that is a surface of revolution obtained by revolving a curve about an axis. Let's call this the x-axis and the rays parallel to this axis are focused at the origin. We will use differential equations to derive the equation for the curve. A cross section of the solar collector showing the curve is pictured: 1. The law of reflection says that the angles y and 8 are equal. Use this and geometry to show that 3 = 20. 2. The slope of the tangent line to the curve at point (r.y) is given by = tano. We can see from the diagram that y/z= tan 3. Show that 2dy/dz 1-(dy/dr) Useful trig identities are sin(28) = 2 sin cos 8 and cos(28) = cos 6-sin²8. 3. Solve for dy/dr to find a differential equation whose solution is the curve defining the cross section of the solar collector. 4. Show that the differential equation you found is a homogeneous equation (Section 2.6). Use an appropriate variable substitution to find a separable equation. 5. Use technology to find a general solution to the differential equation and graph a few curves from the one parameter family of solutions. Attach your work (with a staple and your name!) to this worksheet. 6. Extra Credit: Use technology (or great artistic skills) to draw a 3D representation of the solar collector surface.

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