Line L is tangent to the graph of y= x- (x^2/500) at the point Q

a) find the x coordinate of point Q
B) right an equation for line L
C) suppose the graph above were a hill (measured in feet). There is a 50 foot tree growing vertically at the top of the hill. Does a spot light (a point P, on the x axis along the line L) shine on any part of the tree?

To find the x-coordinate of the point of tangency Q, we can set the equation of the tangent line equal to the equation of the curve and solve for x.

a) Find the x-coordinate of point Q:
Given that the tangent line is tangent to the graph of y = x - (x^2/500), we need to substitute the equation of the curve into the equation of the tangent line.

Let's denote the x-coordinate of point Q as qx. The equation of the tangent line is given by: y - (x - qx) = m(x - qx), where m is the slope of the tangent line.

The slope of the tangent line can be found by taking the derivative of the curve equation with respect to x and evaluating it at the point of tangency. The derivative of y = x - (x^2/500) is dy/dx = 1 - (2x/500). Evaluating the derivative at Q, we have:

1 - (2qx/500) = m

Substituting qx back into the equation, we get:

1 - (2qx/500) = m(x - qx)

Since the tangent line is tangent to the curve at point Q, qx is also a solution to the original curve equation. Substituting qx into y = x - (x^2/500), we get:

qx - (qx^2/500) = qx

Simplifying the equation, we have:

qx^2/500 = 0

This equation holds when qx = 0. Therefore, the x-coordinate of point Q is qx = 0.

b) Write an equation for line L:
Using the point-slope form of a line, we can derive an equation for line L using the point Q(0, 0) and the known slope m.

The equation for line L is:

y - 0 = m(x - 0)
y = mx

c) Does the spot light shine on any part of the tree?
Since the tree is at the top of the hill, its position can be represented by the maximum point on the graph of y = x - (x^2 / 500). To determine if the spot light shines on the tree, we need to find whether there is any intersection between the curve and the line L.

To find the intersection point(s), we need to solve the following system of equations:

y = mx
y = x - (x^2 / 500)

Substituting y = mx into the second equation, we have:

mx = x - (x^2 / 500)

Simplifying the equation, we get:

(x^2 / 500) + (1 - m)x = 0

If this equation has real solutions for x, then the spot light shines on a part of the tree. Otherwise, it does not.

To find the conditions for the equation to have real solutions, we can calculate the discriminant. Denoting the discriminant as D, we can use the quadratic formula:

D = (1 - m)^2 - 4(1/500)(0)
D = (1 - m)^2

For the equation to have real solutions, D must be greater than or equal to zero:

(1 - m)^2 ≥ 0

Since a square is always non-negative, the inequality is always true for any value of m. Therefore, the equation always has real solutions.

In conclusion, the spot light shines on a part of the tree.