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 of the form \( h(t) = at^2 + bt + c \). The time \( t \) at which the maximum height occurs can be found using the formula:
\[ t = -\frac{b}{2a} \]
In the given function \( h(t) = -16t^2 + 32t + 5 \):
- \( a = -16 \)
- \( b = 32 \)
Now, we can plug these values into the vertex formula:
\[ t = -\frac{32}{2 \times -16} = -\frac{32}{-32} = 1 \]
So, the t-shirt will reach its maximum height at \( t = 1 \) second.
Next, we need to find the maximum height by substituting \( t = 1 \) back into the height function:
\[ h(1) = -16(1)^2 + 32(1) + 5 \]
Calculating this:
\[ h(1) = -16(1) + 32(1) + 5 = -16 + 32 + 5 = 16 + 5 = 21 \]
Thus, the maximum height of the t-shirt is 21 feet.
Summarizing the results:
- A t-shirt will reach its maximum height at 1 second.
- The maximum height of a t-shirt will be 21 feet.
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