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

Answer:

The time taken by football to hit the ground is:

                      t=2.3561 sec.

Step-by-step 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:

$x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}$

Here we have:

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

Hence, the solution is:

$t=\dfrac{-(-9)\pm \sqrt{(-9)^2-4\times 4\times (-1)}}{2\times 4}\\\\i.e.\\\\t=\dfrac{9\pm \sqrt{81+16}}{8}\\\\i.e.\\\\t=\dfrac{9\pm \sqrt{97}}{8}\\\\i.e.\\\\t=\dfrac{9+\sqrt{97}}{8},\ t=\dfrac{9-\sqrt{97}}{8}$

Also, we know that:

$\sqrt{97}=9.8489$

This means that:

$t=\dfrac{18.8489}{8}\, t=\dfrac{-0.8489}{8}\\\\i.e.\\\\t=2.3561,\ t=-0.1061$

time can't be negative.

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

           

Answer 2

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