Question

Armando kicks a football into the air the equation f(x)=-6x^2+38x+0.25 models the height of the football from the ground, in feet, with respect to the Time x, In seconds. Us a graph or a table to estimate the time for the ball to return to the ground after being kicked.

Answer 1

To find the time a football takes to hit the ground based on the equation f(x)=-6x^2+38x+0.25, solve for x when f(x)=0 using graphical methods or the quadratic formula. Select the positive root as the time the ball returns to the ground.

Step-by-step explanation:

To estimate the time for the football to return to the ground after being kicked, as represented by the equation f(x)=-6x^2+38x+0.25, we need to find the value of x when f(x)=0. This represents the time in seconds when the ball's height above the ground is zero, meaning it has come back to the ground.

To solve for x, we set the equation to zero and solve the quadratic equation:

• 0 = -6x^2 + 38x + 0.25

This can be done using a graphing calculator or by making a table of values. By plotting the quadratic function, we can visually identify the point where the graph intersects the x-axis, which will give us the time when the football hits the ground. Another method is to use the quadratic formula:

• x = (-b ± √(b^2 - 4ac)) / (2a)

Where a = -6, b = 38, and c = 0.25. After solving, we will get two possible times, and we choose the positive value, which represents the actual time the football will take to return to the ground.

Answer 2

The time for the ball to return to the ground after being kicked is 6.34 seconds.

Step-by-step explanation:

The equation representing the height of the football from the ground (in feet) when Armando kicks the football into the air, with respect to the time x (in seconds) is:

`f(x)=-6x^(2)+38x+0.25`

Now, when the ball returns to the ground after being kicked, the height is 0 feet.

Compute the value of x for f (x) = 0 as follows:

`-6x^(2)+38x+0.25=0\\\\x = ( -38 \pm √(38^2 - 4(-6)(0.25)))/( 2(-6) )\\\\x = ( -38 \pm √(1444 - -6))/( -12 )\\\\x = ( -38 \pm √(1450))/( -12 )\\\\x = ( 19)/( 6 ) + ( 5√(58)\, )/( 12 )\\\\x = 6.33991`

Thus, the time for the ball to return to the ground after being kicked is 6.34 seconds.