- #1
hexag1
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Homework Statement
So here's a question from my textbook 'Calculus: Concepts and Contexts' 2nd ed. by James Stewart. This is section 3.6 # 54
We have Cartesian coordinates set up with an ellipse at [tex] x^2 + 4y^2 = 5 [/tex]
To the right of the ellipse a lamppost (in 2D!) stands at x=3 with unknown height. The lamppost shines a light to the left over the ellipse. The ellipse then casts a shadow. The point at (-5,0) marks where the edge of the shadow crosses the x-axis. The shadow-line is a line tangential to the ellipse running from the lamplight to (-5,0). This is the only given value for the shadow-line. The shadow-line touches the ellipse on the top left quadrant.
The Question: how tall is the lamp?
Implicit differention with respect to x gives:
[tex] 2x + 8y*y' = 0 [/tex]
solving for [tex] y' [/tex] we have: [tex] y' = -x/4y [/tex]
Homework Equations
ellipse : [tex] x^2 + 4y^2 = 5 [/tex]
derivative of ellipse : [tex] 2x + 8y*y' = 0 [/tex]
shadow-line intercept at (-5,0)
The Attempt at a Solution
I find it difficult to see how to proceed. I can find expressions for various elements of the problem, but they all seem to be written in terms of each other with no way to find a number for the height of the lamp.
If I call the point where the shadow line (which is tangential to the ellipse) intercepts the ellipse (j,k) then I find that the height of the lamp is -2j/k