To find the maximum height (or the apex) that the ball reaches, we can use the vertex formula for a parabola given by the equation \( h = -16t^2 + 60t + 0 \). The general formula for the time \( t \) at which the maximum height occurs in a quadratic equation \( ax^2 + bx + c \) is given by:
\[ t = -\frac{b}{2a} \]
In this case, \( a = -16 \) and \( b = 60 \). Plugging in the values:
\[ t = -\frac{60}{2(-16)} = \frac{60}{32} = 1.875 \text{ seconds} \]
Next, we substitute \( t = 1.875 \) back into the height equation to find the maximum height:
\[ h = -16(1.875)^2 + 60(1.875) + 0 \]
Calculating \( (1.875)^2 \):
\[ (1.875)^2 = 3.515625 \]
Now, substituting this value into the equation:
\[ h = -16(3.515625) + 60(1.875) \]
Calculating each term:
\[ -16(3.515625) = -56.25 \] \[ 60(1.875) = 112.5 \]
Combining the terms:
\[ h = -56.25 + 112.5 = 56.25 \]
Thus, the maximum height the ball will reach is 56.25 feet.
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