Question

1. a track and field athlete whose event is the shot put releases a shot. when the shot is released at an angle of 30°, its height f(x), in feet, can be modeled by f(x)= - 0.01x² + 0.6x + 6.3 where x is the shots horizontal distance, in feet, from its point of release. (the graph is for reference only; use the function for all calculations. round to the tenths place, as needed.) reference hw 3.1, #13 a. what is the maximum height of the shot put, and how far from its point of release does this occur? b. what is the shots maximum horizontal distance, or the distance of the throw? c. from what height was the shot released?

Explanation:

Step1: Identify the quadratic - function form

The given function is $f(x)=-0.01x^{2}+0.6x + 6.3$, which is in the form $y = ax^{2}+bx + c$ where $a=-0.01$, $b = 0.6$, and $c = 6.3$.

Step2: Find the x - coordinate of the vertex

The x - coordinate of the vertex of a quadratic function $y = ax^{2}+bx + c$ is given by $x=-\frac{b}{2a}$. Substitute $a=-0.01$ and $b = 0.6$ into the formula:
$x=-\frac{0.6}{2\times(-0.01)}=\frac{0.6}{0.02}=30$.

Step3: Find the maximum height (y - coordinate of the vertex)

Substitute $x = 30$ into the function $f(x)=-0.01x^{2}+0.6x + 6.3$:
$f(30)=-0.01\times(30)^{2}+0.6\times30 + 6.3=-0.01\times900 + 18+6.3=-9 + 18+6.3=15.3$.

Step4: Find the maximum horizontal distance (x - intercepts)

Set $f(x)=0$, so $-0.01x^{2}+0.6x + 6.3 = 0$. Multiply through by - 100 to get $x^{2}-60x - 630=0$.
Use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$, where $a = 1$, $b=-60$, and $c=-630$.
$x=\frac{60\pm\sqrt{(-60)^{2}-4\times1\times(-630)}}{2\times1}=\frac{60\pm\sqrt{3600 + 2520}}{2}=\frac{60\pm\sqrt{6120}}{2}=\frac{60\pm78.23}{2}$.
We take the positive root $x=\frac{60 + 78.23}{2}\approx69.1$.

Step5: Find the initial height

Set $x = 0$ in the function $f(x)=-0.01x^{2}+0.6x + 6.3$. Then $f(0)=6.3$.

Answer:

a. The maximum height is 15.3 feet and it occurs 30 feet from the point of release.
b. The maximum horizontal distance is approximately 69.1 feet.
c. The shot was released from a height of 6.3 feet.