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?
they drew a diagram for it and Q is actually on the left side of the graph making the line L an upward slant, and yes i meant "write"
a) To find the x-coordinate of point Q, we need to find the point where the line is tangent to the graph.
For a line to be tangent to a curve, its slope should be equal to the slope of the curve at that point.
The given equation is y = x - (x^2/500).
To find the slope of the curve at any point, we need to find the derivative of the equation with respect to x.
dy/dx = 1 - (2x/500)
For the line L to be tangent, its slope should be equal to the slope of the curve at point Q.
Therefore, we can set the slope of the line L equal to dy/dx and solve for x.
1 - (2x/500) = 1
Simplifying the equation, we get:
- (2x/500) = 0
Dividing both sides by -2/500, we have:
x = 0
So, the x-coordinate of point Q is 0.
b) To write an equation for line L, we need the slope of the line and the coordinates of point Q.
From the previous step, we found that the x-coordinate of point Q is 0.
Since the line is tangent to the graph at point Q, its slope is equal to the slope of the graph at that point, which is dy/dx = 1 - (2x/500).
Plugging in the x-coordinate of point Q, we get:
m = 1 - (2(0)/500)
m = 1 - 0
m = 1
The slope of the line is 1 and the coordinates of point Q are (0, f(0)), so the equation of line L is:
y - f(0) = m(x - 0)
y - f(0) = x
c) To determine if the spot light shines on any part of the 50-foot tree, we need to check at what point the height of the hill and the height of the tree intersect.
Since the height of the hill is given by the equation y = x - (x^2/500), and the height of the tree is fixed at 50 feet, we need to find the x-coordinate where y = 50.
Setting y = 50 in the equation of the hill, we have:
50 = x - (x^2/500)
Rearranging the equation, we get:
x^2 - 500x + 25000 = 0
Solving this quadratic equation, we find that there are two x-values where the height of the hill equals 50.
Therefore, there are two spots along the x-axis (line L) where the spot light shines on the tree.
a) To find the x-coordinate of point Q, we need to find the value of x where the line L is tangent to the graph of y = x - (x^2/500).
To do this, we need to find the derivative of the function y = x - (x^2/500) and solve for x when the derivative is equal to the slope of L.
First, let's find the derivative:
dy/dx = 1 - (2x/500)
dy/dx = 1 - (x/250)
To find the slope of the tangent line L, we equate the derivative to the slope of L. Let's assume the slope of L is m.
1 - (x/250) = m
Next, we substitute the equation of the graph y = x - (x^2/500) into the equation of the tangent line L:
m = 1 - (x/250)
m = x - (x^2/500)
Now, we have a system of equations, where the first equation represents the slope of L and the second equation represents the graph of y = x - (x^2/500). We can solve these equations simultaneously to find the x-coordinate of point Q.
b) To find the equation for line L, we need to have its slope and a point on the line. We can use the x-coordinate of point Q and the derivative equation we found earlier to calculate the slope, and then substitute the x-coordinate and the slope into the equation of a line to obtain the equation of line L.
c) To determine if the spot light shines on any part of the tree, we need to find the y-coordinate of the tree and see if it intersects with the line L. If the y-coordinate of the tree is greater than the y-coordinate of the point P on line L, then the spot light will shine on the tree.
Did you want that tangent to be horizontal ?
if so, then
dy/dx = 1 - x/250
to be horizontal tangent, dy/dx = 0
x/250 = 1
x = 250
then y = 250 - (250^2)/500 = 125
Q is (250,125)
b) Did you mean" write" an equation for line L ?
Until you clarify your question, I will stop here.