An athlete whose event is the shot put releases a shot. When the shot whose path is shown by the graph to the right is released at an angle of $35^{\circ}$, its height, $\mathrm{f}(\mathrm{x})$, in feet, can be modeled by $f(x)=-0.01 x^{2}+0.7 x+5.9$, where $\mathrm{x}$ is the shot's horizontal distance, in feet, from its point of release. Use this model to solve parts (a) through (c) and verify your answers using the graph.
a. What is the maximum height of the shot and how far from its point of release does this occur?
The maximum height is 18.15 , which occurs 35 feet from the point of release.
(Type an integer or decimal rounded to four decimal places as needed)
b. What is the shot's maximum horizontal distance, to the nearest tenth of a foot, or the distance of the throw?
77.6 - feet
(Type an integer or decimal rounded to the nearest tenth as needed.)
c. From what height was the shot released?
furet
(Type an integer or decimal rounded to the nearest tenth as needed.)

Step 1 ：Given the quadratic function \(f(x) = -0.01x^2 + 0.7x + 5.9\), we can find the maximum height, maximum horizontal distance, and the height from which the shot was released.

Step 2 ：To find the maximum height, we first find the vertex of the parabola. The x-coordinate of the vertex is given by \(x_v = \frac{-b}{2a}\), where \(a = -0.01\) and \(b = 0.7\).

Step 3 ：\(x_v = \frac{-0.7}{2(-0.01)} = 35\)

Step 4 ：Substitute \(x_v\) into the function to find the maximum height: \(f(35) = -0.01(35)^2 + 0.7(35) + 5.9 = 18.15\)

Step 5 ：\(\boxed{\text{a. The maximum height of the shot is 18.15 feet, which occurs 35 feet from the point of release.}}\)

Step 6 ：To find the maximum horizontal distance, we need to find the x-intercepts of the function. Set \(f(x) = 0\) and solve for \(x\):

Step 7 ：\(-0.01x^2 + 0.7x + 5.9 = 0\)

Step 8 ：Using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = -0.01\), \(b = 0.7\), and \(c = 5.9\).

Step 9 ：\(x = \frac{-0.7 \pm \sqrt{(0.7)^2 - 4(-0.01)(5.9)}}{2(-0.01)}\)

Step 10 ：\(x \approx 0, 77.6\)

Step 11 ：\(\boxed{\text{b. The shot's maximum horizontal distance is 77.6 feet (rounded to the nearest tenth).}}\)

Step 12 ：The height from which the shot was released is the value of the function at \(x = 0\): \(f(0) = -0.01(0)^2 + 0.7(0) + 5.9 = 5.9\)

Step 13 ：\(\boxed{\text{c. The height from which the shot was released is 5.9 feet.}}\)