SOLUTION: Hiroshi is trying out for the position of kicker on the football team. He wants to know at what angle he should kick the ball for maximum distance. He has used a machine that kicks

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Question 1200324: Hiroshi is trying out for the position of kicker on the football team. He wants to know at what angle he should kick the ball for maximum distance. He has used a machine that kicks footballs with constant velocity but at varying angles. Hiroshi has collected some data and used quadratic regression on his graphing calculator to determine that the relation between angle and distance is given by the equation d=-0.1a^2 + 8.5a - 40 where a is the angle in degrees, and d is the distance in metres. a) Determine the zeros of this graph. b) Which angle gives the maximum distance, and what is t he maximum distance?
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Part (a)

I'll use x in place of the variable 'a'.
I'll also use y in place of d.

That means
d = -0.1a^2 + 8.5a - 40
will be updated to
y = -0.1x^2 + 8.5x - 40

x = angle in degrees
y = distance in meters

The term "zeros" is another term for "roots" and "x intercept".
These are the values of x that make y = 0.
y = -0.1x^2 + 8.5x - 40
0 = -0.1x^2 + 8.5x - 40
-0.1x^2 + 8.5x - 40 = 0

That last equation is in the form
ax^2+bx+c = 0
where in this case
a = -0.1
b = 8.5
c = -40
Plug those values into the quadratic formula.
x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29

x+=+%28-8.5%2B-sqrt%28%288.5%29%5E2-4%28-0.1%29%28-40%29%29%29%2F%282%28-0.1%29%29

x+=+%28-8.5%2B-sqrt%2856.25%29%29%2F%28-0.2%29

x+=+%28-8.5%2B-++7.5%29%2F%28-0.2%29

x+=+%28-8.5%2B7.5%29%2F%28-0.2%29 or x+=+%28-8.5-7.5%29%2F%28-0.2%29

x+=+%28-1%29%2F%28-0.2%29 or x+=+%28-16%29%2F%28-0.2%29

x+=+5 or x+=+80
Now recall that x represents the angle.
An angle of x = 5 degrees causes the distance to be zero. The same applies to an angle of x = 80 degrees.
You can verify through the use of a graph.

Side note: This green parabola is NOT the path the ball takes even if it might seem so.

Desmos is a useful graphing tool.
https://www.desmos.com/calculator/bcyu5az98d
That link leads to the interactive graph so you can move the window around, zoom in/out, etc.
To determine a particular root's location in Desmos, simply click on the root in question.
It should reveal the coordinates. You may have to click twice.

Summary:
The roots are a = 5 and a = 80
This is where the green parabola crosses the x axis.

==============================================================================================================================

Part (b)

Refer to the graph above.
The highest point is the vertex.
Due to symmetry of the parabola, the x coordinate of the vertex is at the midpoint of the roots.
The vertex is located on the axis of symmetry.

We found the roots to be x = 5 and x = 80.
The midpoint has us add them up and divide by 2
(5+80)/2 = 85/2 = 42.5
The axis of symmetry is the vertical line x = 42.5
This is the x coordinate of the vertex.

Plug that into the original equation to find the y coordinate of the vertex
y = -0.1x^2 + 8.5x - 40
y = -0.1(42.5)^2 + 8.5(42.5) - 40
y = 140.625

The vertex is located at (x,y) = (42.5, 140.625) which is the highest point on the green parabola.
It tells us that an angle of a = 42.5 degrees leads to the distance to max out at d = 140.625 meters

Let's update the graph to add the vertex.

To determine the vertex's location in Desmos, simply click on the highest point.
It should reveal the coordinates. You may have to click twice.

Summary:
Angle that maxes distance is 42.5 degrees
That distance being 140.625 meters