Question

Armando kicks a football into the air. The function f(x) = - 7x² + 38x+0.22 models the height of the football from the ground, in feet, with respect to the time x inseconds. Use a graph or table to estimate the time for the ball to return to the ground after being kicked

Answer 1

Hi there. To solve this question, we'll have to remember some properties about parabolas and its roots.

Given the function:

`f(x)=-7x^2+38x+0.22`

that models the height of the football Armando kicked from the ground, in feet, with respect to time x in seconds, we have to determine:

The time it takes for the ball to return to the ground after being kicked.

Let's suppose that the ball was kicked from x = 0, since the times x is given in seconds and we can't have negative time.

Before supposing it, in fact we have to determine which moments the ball was at the ground, that is, f(x) = 0. We're finding the roots of the function:

`\begin{gathered} -7x^2+38x+0.22=0 \\ \end{gathered}`

To solve this quadratic equation, remember the general solution to a quadratic equation

`ax^2+bx+c=0,a\text{ not equal to 0.}`

Is given by the formula:

`(-b\pm√(b^2-4ac))/(2a)`

Plugging a = -7, b = 38 and c = 0.22, we get:

`x=(-38\pm√(38^2-4\cdot7\cdot0.22))/(2\cdot(-7))`

Multiply the values, square the number and add inside the radical.

`x=(-38\pm√(1444-6.16))/(-14)=(-38\pm√(1437.84))/(-14)`

Separing the solutions and calculating their values, we get

`\begin{gathered} x=(-38\pm37.92)/(-14) \\ \\ x_1=(-38+37.92)/(-14)=0.006 \\ \\ x_2=(-38-37.92)/(-14)=5.423 \end{gathered}`

In this case, we before had supposed the ball started from x = 0. This is not really necessary because we found that the roots of this functions are contained in the positive x-axis.

To find the time for the ball to return to the ground, we make:

`|x_2-x_1|=|5.423-0.006|=|5.417|=5.417`

Or 5.42 seconds.

Graphically, what we have is: