Answer:
Part a) About 48.6 feet
Part b) About 8.3 feet
Part c) The domain is [tex]0 \leq x \leq 48.6\ ft[/tex] and the range is [tex]0 \leq y \leq 8.3\ ft[/tex]
Step-by-step explanation:
we have
[tex]y=-0.014x^{2} +0.68x[/tex]
This is a vertical parabola open downward (the leading coefficient is negative)
The vertex represent a maximum
where
x is the ball's distance from the catapult in feet
y is the flight of the balls in feet
Part a) How far did the ball fly?
Find the x-intercepts or the roots of the quadratic equation
Remember that
The x-intercept is the value of x when the value of y is equal to zero
The formula to solve a quadratic equation of the form
[tex]ax^{2} +bx+c=0[/tex]
is equal to
[tex]x=\frac{-b(+/-)\sqrt{b^{2}-4ac}} {2a}[/tex]
in this problem we have
[tex]-0.014x^{2} +0.68x=0[/tex]
so
[tex]a=-0.014\\b=0.68\\c=0[/tex]
substitute in the formula
[tex]x=\frac{-0.68(+/-)\sqrt{0.68^{2}-4(-0.014)(0)}} {2(-0.014)}[/tex]
[tex]x=\frac{-0.68(+/-)0.68} {(-0.028)}[/tex]
[tex]x=\frac{-0.68(+)0.68} {(-0.028)}=0[/tex]
[tex]x=\frac{-0.68(-)0.68} {(-0.028)}=48.6\ ft[/tex]
therefore
The ball flew about 48.6 feet
Part b) How high above the ground did the ball fly?
Find the maximum (vertex)
[tex]y=-0.014x^{2} +0.68x[/tex]
Find out the derivative and equate to zero
[tex]0=-0.028x +0.68[/tex]
Solve for x
[tex]0.028x=0.68[/tex]
[tex]x=24.3[/tex]
Alternative method
To determine the x-coordinate of the vertex, find out the midpoint between the x-intercepts
[tex]x=(0+48.6)/2=24.3\ ft[/tex]
To determine the y-coordinate of the vertex substitute the value of x in the quadratic equation and solve for y
[tex]y=-0.014(24.3)^{2} +0.68(24.3)[/tex]
[tex]y=8.3\ ft[/tex]
the vertex is the point (24.3,8.3)
therefore
The ball flew above the ground about 8.3 feet
Part c) What is a reasonable domain and range for this function?
we know that
A reasonable domain is the distance between the two x-intercepts
so
[tex]0 \leq x \leq 48.6\ ft[/tex]
All real numbers greater than or equal to 0 feet and less than or equal to 48.6 feet
A reasonable range is all real numbers greater than or equal to zero and less than or equal to the y-coordinate of the vertex
so
we have the interval -----> [0,8.3]
[tex]0 \leq y \leq 8.3\ ft[/tex]
All real numbers greater than or equal to 0 feet and less than or equal to 8.3 feet
