Example

Finding the Maximum Height of a Stone Modeled by h=16t2+128t+32h = -16t^2 + 128t + 32

To find the maximum height of a stone thrown upward from a height of 32 feet at a rate of 128 ft/sec, modeled by the quadratic equation h=16t2+128t+32h = -16t^2 + 128t + 32, we can find the vertex of the corresponding downward-opening parabola. Because the leading coefficient a=16a = -16 is negative, the parabola opens downward, meaning the vertex represents the maximum value.

First, determine the time tt at which the maximum height occurs by calculating the axis of symmetry: t=b2at = -\frac{b}{2a} t=1282(16)t = -\frac{128}{2(-16)} t=4t = 4

The stone reaches its maximum height after 4 seconds. Next, evaluate the maximum height by substituting t=4t = 4 into the equation: h=16(4)2+128(4)+32h = -16(4)^2 + 128(4) + 32 h=256+512+32h = -256 + 512 + 32 h=288h = 288

The maximum height of the stone is 288 feet.

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Updated 2026-04-21

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