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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.

Verified Answer :

AEquation 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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adioabiola

The answers to the questions are as follows;

  • a) The x-coordinate of the point Q is; 100ft
  • b) The equation of for the line l as described is; y = 3/5x + 20
  • c) The tangent line goes through (250,170), and hence, the top 5 feet of the tree would be above the tangent line, in the glow of the spotlight.

Tangent line to Curves

Part A:

Since, the line l is tangent to the curve at point Q; it follows that the equation of the line l is given as the differential of the equation of the curve.

Line l equation: y = mx + 20

  • where m = y' = 1- (1/250)x

At point Q, Equation of the line, l equals the equation of the parabola.

  • (1-x/250)x + 20 = x - x^2/500

  • 20 = x^2/250 - x^2/500 = x^2/500

  • x = √(20*500) = 100ft

Part B:

By substituting x = 100 into the equation; slope, m is given as;

  • m = 1 - 100/250 = 3/5.

Ultimately, Equation of line L is: y = 3/5x + 20

Part C:

If a spotlight is located at the x-intercept of line L given by coordinates (-100/3,0), it follows that it's light goes only above line L past the point where line L is tangent to the hill at (100,80).

Ultimately, the highest point on the equation of y = x - x²/500 is at (250,125).

On this note, a tree 50 feet tall at that location would reach up to (250,175). The tangent line therefore goes through (250,170), that the top 5 feet of the tree would be above the given tangent line, in the glow of the spotlight.

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