To find the time it takes for the t-shirt to hit the ground using the function \( h(t) = -16t^2 + 32t + 5 \), we need to determine when the height \( h(t) \) equals zero.
This sets up the equation:
\[ -16t^2 + 32t + 5 = 0 \]
We can apply the quadratic formula, which is given by:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our equation, the coefficients are as follows:
- \( a = -16 \)
- \( b = 32 \)
- \( c = 5 \)
Let's calculate the discriminant (\( b^2 - 4ac \)) first:
\[ b^2 = 32^2 = 1024 \] \[ 4ac = 4 \times (-16) \times 5 = -320 \] \[ b^2 - 4ac = 1024 - (-320) = 1024 + 320 = 1344 \]
Now we can apply the quadratic formula:
\[ t = \frac{-32 \pm \sqrt{1344}}{2 \times -16} \]
Calculating \( \sqrt{1344} \):
\[ \sqrt{1344} \approx 36.66 \]
Now, substituting back into the formula:
\[ t = \frac{-32 \pm 36.66}{-32} \]
This will yield two possible solutions for \( t \):
-
\( t = \frac{-32 + 36.66}{-32} = \frac{4.66}{-32} \) \[ t \approx -0.1456 \quad (\text{not a valid time since it's negative}) \]
-
\( t = \frac{-32 - 36.66}{-32} = \frac{-68.66}{-32} \) \[ t \approx 2.144375 \]
Rounding to the nearest hundredth gives:
\[ t \approx 2.14 \text{ seconds} \]
Thus, the t-shirt will hit the ground at approximately 2.14 seconds.
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