a football player punts a football at an inclination of 45 degrees to the horizontal at an initial velocity…

a football player punts a football at an inclination of 45 degrees to the horizontal at an initial velocity of 80 feet per second. the height of the football in feet is given by the quadratic function h(x)= - 32x^2/(80)^2 + x + 4, where x is the horizontal distance of the football from the point at which it was kicked. what was the horizontal distance of the football from the point at which it was kicked when the height of the ball is at a maximum? what is the maximum height of the football?

a football player punts a football at an inclination of 45 degrees to the horizontal at an initial velocity of 80 feet per second. the height of the football in feet is given by the quadratic function h(x)= - 32x^2/(80)^2 + x + 4, where x is the horizontal distance of the football from the point at which it was kicked. what was the horizontal distance of the football from the point at which it was kicked when the height of the ball is at a maximum? what is the maximum height of the football?

Answer

Explanation:

Step1: Identify the quadratic - function form

The given function is $h(x)=-\frac{32x^{2}}{80^{2}}+x + 4=-\frac{32x^{2}}{6400}+x + 4=-\frac{1}{200}x^{2}+x + 4$, which is in the form $y = ax^{2}+bx + c$ where $a=-\frac{1}{200}$, $b = 1$, and $c = 4$.

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 $a=-\frac{1}{200}$ and $b = 1$ into the formula: $x=-\frac{1}{2\times(-\frac{1}{200})}=\frac{1}{\frac{1}{100}} = 100$ feet. This is the horizontal distance when the height is maximum.

Step3: Find the maximum height

Substitute $x = 100$ into the function $h(x)=-\frac{1}{200}x^{2}+x + 4$. $h(100)=-\frac{1}{200}\times(100)^{2}+100 + 4$. First, calculate $-\frac{1}{200}\times(100)^{2}=-\frac{10000}{200}=-50$. Then, $h(100)=-50 + 100+4=54$ feet.

Answer:

The horizontal distance when the height is maximum is 100 feet. The maximum height of the football is 54 feet.