a punter kicked the football into the air with an upward velocity of 62 ft/s. Its height h in feet after t seconds is given by the function h= -16t ^2 + 62 t + 2. How long does it take for the ball to hit the ground?
16
To find out how long it takes for the ball to hit the ground, we need to determine the value of t when the height, h, is equal to 0.
In the given function, h = -16t^2 + 62t + 2. By setting h = 0, we get:
0 = -16t^2 + 62t + 2
Now we have a quadratic equation, which can be solved using various methods, such as factoring, completing the square, or using the quadratic formula. Let's solve it using the quadratic formula.
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = -16, b = 62, and c = 2. Plugging these values into the formula, we get:
t = (-62 ± √(62^2 - 4(-16)(2))) / (2(-16))
Now, let's simplify it further:
t = (-62 ± √(3844 + 128)) / (-32)
t = (-62 ± √(3972)) / (-32)
t = (-62 ± 62.98) / (-32)
We have two possible solutions:
1) t = (-62 + 62.98) / (-32)
t ≈ 0.02 / (-32)
t ≈ -0.00063 seconds (approx.)
2) t = (-62 - 62.98) / (-32)
t ≈ -124.98 / (-32)
t ≈ 3.90 seconds (approx.)
Since time cannot be negative in this context, we discard the first solution. Therefore, it takes approximately 3.90 seconds for the ball to hit the ground.