To find the time it takes for the football to hit the ground, we need to solve the equation \( h = -16t^2 + 6t + 4 \) for \( h = 0 \) (when the football hits the ground).
We set up the equation:
\[ -16t^2 + 6t + 4 = 0 \]
Now, we can use the quadratic formula, which is:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our equation, \( a = -16 \), \( b = 6 \), and \( c = 4 \).
First, we calculate the discriminant:
\[ b^2 - 4ac = 6^2 - 4(-16)(4) = 36 + 256 = 292 \]
Now we apply the quadratic formula:
\[ t = \frac{-6 \pm \sqrt{292}}{2(-16)} \]
Calculating \( \sqrt{292} \):
\[ \sqrt{292} \approx 17.09 \]
Now substituting back into the quadratic formula:
\[ t = \frac{-6 \pm 17.09}{-32} \]
This gives us two possible solutions for \( t \):
\[ t = \frac{-6 + 17.09}{-32} = \frac{11.09}{-32} \approx -0.346 \]
(This value is negative and thus not a valid solution since time cannot be negative.)
\[ t = \frac{-6 - 17.09}{-32} = \frac{-23.09}{-32} \approx 0.722 \]
We round \( 0.722 \) to the nearest hundredth:
\[ t \approx 0.72 \]
Thus, the time it takes for the football to hit the ground is 0.72 seconds.
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