Question

If a football player passes a football from 4 feet off the ground with an initial velocity of 36 feet per second, how long will it take the football to hit the ground? Use the equation h = −16t2 + 36t + 4.

Answer 1

0 = -16t² + 36t + 4
0 = -8t² + 18t + 2 ← Complete the square to solve
-8(t² - 9t/2) = -2
8(t² - 9t/4 + 81/64) = 2 + 81/8
8(t - 9/8)² = 97/8
(t - 9/8)² = 97/64
t - (t = 0)
(h = 0) at:
t = (9 + √97)/8 ≅ 2.356 seconds

Answer 2

The time taken by football to hit the ground is:

t=2.3561 sec.

Explanation:

The height(h) of the football at a time t is given by the equation:

`h=-16t^2+36t+4`

Now, we are asked to find the time the ball will take to hit the ground.

i.e. we are asked to find the value of t when h=0

i.e.

`-16t^2+36t+4=0\\\\i.e.\\\\16t^2-36t-4=0`

( Since, on multiplying both the sides of the equation by -1 )

Now, we divide both the side of the equation by 4 to get:

`4t^2-9t-1=0`

We know that the solution of the quadratic equation of the type:

`ax^2+bx+c=0`

is given by the formula:

`x=(-b\pm √(b^2-4ac))/(2a)`

Here we have:

`a=4,\ b=-9\ and\ c=-1`

Hence, the solution is:

`t=(-(-9)\pm √((-9)^2-4* 4* (-1)))/(2* 4)\\\\i.e.\\\\t=(9\pm √(81+16))/(8)\\\\i.e.\\\\t=(9\pm √(97))/(8)\\\\i.e.\\\\t=(9+√(97))/(8),\ t=(9-√(97))/(8)`

Also, we know that:

`√(97)=9.8489`

This means that:

`t=(18.8489)/(8)\, t=(-0.8489)/(8)\\\\i.e.\\\\t=2.3561,\ t=-0.1061`

time can't be negative.

Hence, the solution is:
`t=2.3561\ sec.`