Line l is tangent to the graph of y= x - x²/500 at point q & it crosses the y-axis at (0, 20)
a). find the x-coordinate of point q b. write an equation for line l c) suppose the graph of y=x-x^2/500, where x and y are measured in feet, represents a hill. there is a 50-foot tree growing vertically at the top of the hill. does a spotlight at point p (point p is where line l crosses the x-axis) directed along the line l shine on any part of the tree? show the work that leads to your conclusion.

A. Equation of line l: y = mx + 20 where m = y' = 1- (1/250)x 

At the point Q, The equation of the line equals the equation of the parabola. 

So (1-x/250)x + 20 = x - x^2/500 

20 = x^2/250 - x^2/500 = x^2/500 
x = sqrt(20*500) = 100ft 

B. m = 1 - 100/250 = 3/5. 

Equation of line L is y = 3/5x + 20 

C)  If a spotlight is located at the x intercept of line L (-33 1/3,0), its light will go only above line L past the point where line L is tangent to the hill at (100,80) because the hill blocks the light below the tangent line. 

The highest point on the equation of y = x - x²/500 is at (250,125), so a tree 50 feet tall at that location would reach up to (250,175). The tangent line goes through (250,170), so the top 5 feet of the tree would be above the tangent line, in the glow of the spotlight. 



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