To find the time it takes for the t-shirt to reach its maximum height, we can use the vertex formula for a quadratic function given in the form \( h = at^2 + bt + c \). The time \( t \) at which the maximum height occurs is given by the formula:
\[ t = -\frac{b}{2a} \]
In our function, \( h = -16t^2 + 72t + 5 \):
- The coefficient \( a = -16 \)
- The coefficient \( b = 72 \)
Now, substituting \( a \) and \( b \) into the vertex formula:
\[ t = -\frac{72}{2 \cdot (-16)} = -\frac{72}{-32} = \frac{72}{32} = 2.25 \text{ seconds} \]
Next, we calculate the maximum height by substituting \( t = 2.25 \) back into the height function \( h \):
\[ h = -16(2.25)^2 + 72(2.25) + 5 \]
Calculating \( 2.25^2 \):
\[ 2.25^2 = 5.0625 \]
Now substituting into the height equation:
\[ h = -16(5.0625) + 72(2.25) + 5 \] \[ = -81 + 162 + 5 \] \[ = 86 \text{ feet} \]
Thus, we have:
- A t-shirt will reach its maximum height at 2.25 seconds.
- The maximum height of a t-shirt will be at 86 feet.
Now, to determine when the t-shirt hits the ground, we set the height \( h \) to 0 and solve the equation:
\[ -16t^2 + 72t + 5 = 0 \]
Using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- \( a = -16 \)
- \( b = 72 \)
- \( c = 5 \)
Calculating the discriminant:
\[ b^2 - 4ac = 72^2 - 4(-16)(5) = 5184 + 320 = 5504 \]
Now applying the quadratic formula:
\[ t = \frac{-72 \pm \sqrt{5504}}{2(-16)} \] Calculating \( \sqrt{5504} \):
\[ \sqrt{5504} \approx 74.24 \]
Now substituting back:
\[ t = \frac{-72 \pm 74.24}{-32} \]
Calculating the two possible values for \( t \):
- \( t = \frac{-72 + 74.24}{-32} = \frac{2.24}{-32} \) (this will yield a negative time, hence not valid)
- \( t = \frac{-72 - 74.24}{-32} = \frac{-146.24}{-32} = 4.57 \)
Thus, rounding to the nearest hundredth, we have:
A t-shirt will hit the ground in about 4.57 seconds.
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