The answer is: The maximum height of the arrow is 125 feet.
To solve this problem, we need to know the motion of the arrow is described by a quadratic equation (parabola). Also, we are given a parabola with a negative coefficient. So, to solve this problem, we need to remember the following:
- To know if the parabola is opening upward or downward, we need to look the sign of the coefficient of the quadratic term.
- We need to remember that the highest or lowest point of a parabola is given by the y-coordinate of the vertex.
We can calculate the vertex using the following formula:
[tex]x=\frac{-b}{2a}[/tex]
Then, to find the y-coordinate of the vertex, we need to substitute the x value into the parabola equation.
So, we are given the pathway of the arrow represented by the quadratic equation:
[tex]h=-16t^{2} +80t+25[/tex]
[tex]f(t)=-16t^{2} +80t+25[/tex]
[tex]y=-16t^{2} +80t+25[/tex]
Where,
[tex]a=-16\\b=80\\c=25[/tex]
So, calculating the vertex coordinates in order to find the maximum height of the arrow (highest point), we have:
[tex]t=\frac{-b}{2a}=\frac{-80}{2(-16)}=\frac{-80}{-32}=2.5[/tex]
Then, substituting "t" into the equation of the parabola, we have:
[tex]y=-16(2.5)^{2} +80(2.5)+25[/tex]
[tex]y=-16*6.25 +200+25[/tex]
[tex]y=-100 +200+25=125[/tex]
Therefore, we have that the vertex coordinates are (2.5,125) being the y-coordinate the highest point of the parabola, which is equal to the maximum height of the arrow.
Hence, the maximum height of the arrow is 125 feet.
Have a nice day!
Have a nice day!
