Question

Armando kicks a football into the air. The function f(x)= -6x² + 39x+0.27 models the height of the football from the ground, in feet, with respect to the time x in seconds. 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)= -6x^2 + 39x+0.27`

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:

`-6x^2 + 39x+0.27=0`

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

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

Is given by the formula:

`(-b\pm√(b^2-4ac) )/(2a)`

Plugging a = -6, b = 39 and c = 0.27, we get:

`x=(-39\pm√(39^2-4*6*0.27) )/(2*(-6))`

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

`x=(-39\pm√(1521-6.48) )/(-12) =(-39\pm√(1514.52) )/(-12)`

Separating the solutions and calculating their values, we get

`x=(-39\pm38.92)/(-12)`

`x_1=(-39-38.92)/(-12)=0.007`

`x_2=(-39+38.92)/(-12)=6.493`

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|=|6.493-0.007|=|6.486|=6.486`

Or 6.49 seconds.