Question

a golf ball is hit with an initial velocity of 140 feet per second at an inclination of 45° to the horizontal. in physics, it is established that the height h of the golf ball is given by the function h(x)=\frac{-32x^{2}}{140^{2}}+x, where x is the horizontal distance that the golf ball has traveled. complete parts (a) through (g). (f) use a graphing utility to determine the distance that the ball has traveled when the height of the ball is 100 feet. choose the correct answer below and, if necessary, fill in the answer box to complete your choice. a. the ball has traveled 125.86,486.64 feet. (use a comma to separate answers as needed. round to two decimal places as needed.) b. the ball has not traveled. (g) create a table with tblstart = 0 and δtbl = 25. to the nearest 25 feet, how far does the ball travel before it reaches a maximum height? what is the maximum height? to the nearest 25 feet, the ball travels 300 feet. the maximum height is feet. (round to two decimal places as needed.)

Explanation:

Step1: Set up the equation

We are given the height - function $h(x)=\frac{-32x^{2}}{140^{2}}+x$ and we want to find the maximum of this function. The function $h(x)$ is a quadratic function in the form $y = ax^{2}+bx + c$, where $a=\frac{-32}{140^{2}}$ and $b = 1$, $c = 0$.

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 $x=-\frac{b}{2a}$. Substituting $a=\frac{-32}{140^{2}}$ and $b = 1$ into the formula:

$$
LATEXBLOCK0
$$

Rounding to the nearest 25 feet, $x\approx300$ feet.

Step3: Find the maximum height

Substitute $x = 306.25$ into the function $h(x)=\frac{-32x^{2}}{140^{2}}+x$.

$$
LATEXBLOCK1
$$

Answer:

The ball travels 300 feet (to the nearest 25 feet) and the maximum height is 153.13 feet.