The function given is \( y = -4t^2 + 10t \), which represents the height \( y \) of an object at time \( t \). This is a quadratic function in the standard form \( y = ax^2 + bx + c \), where \( a = -4 \), \( b = 10 \), and \( c = 0 \).
To find the maximum height of the object, we can use the vertex formula for a quadratic function, which occurs at \( t = -\frac{b}{2a} \).
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Calculate \( t \): \[ t = -\frac{b}{2a} = -\frac{10}{2 \times -4} = \frac{10}{8} = 1.25 \text{ seconds} \]
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Substitute \( t = 1.25 \) into the height function to find the maximum height: \[ y = -4(1.25)^2 + 10(1.25) \] \[ = -4(1.5625) + 12.5 \] \[ = -6.25 + 12.5 \] \[ = 6.25 \text{ feet} \]
Therefore, the maximum height the object will reach is 6.25 feet.
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