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Advanced Math / Nonlinear functions Difficulty: Medium

$h (t) = - 16 t^{2} + b$

The function $h$ estimates an object’s height, in feet, above the ground $t$ seconds after the object is dropped, where $b$ is a constant. The function estimates that the object is $3,364$ feet above the ground when it is dropped at $t equals 0$ . Approximately how many seconds after being dropped does the function estimate the object will hit the ground?

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Explanation

Choice B is correct. It's given that the function $h$ estimates that the object is $3,364$ feet above the ground when it's dropped at $t equals 0$ . Substituting $3,364$ for $h (t)$ and $0$ for $t$ in the function $h$ yields $3,364 = - 16 {(0)}^{2} + b$ , or $3,364 equals b$ . Substituting $3,364$ for $b$ in the function $h$ yields $h (t) = - 16 t^{2} + 3,364$ . When the object hits the ground, its height will be $0$ feet above the ground. Substituting $0$ for $h (t)$ in $h (t) = - 16 t^{2} + 3,364$ yields $0 = - 16 t^{2} + 3,364$ . Adding $16 t squared$ to each side of this equation yields $16 t squared equals 3,364$ . Dividing each side of this equation by $16$ yields $t squared equals 210.25$ . Since the object will hit the ground at a positive number of seconds after it's dropped, the value of $t$ can be found by taking the positive square root of each side of this equation, which yields $t equals 14.50$ . It follows that the function estimates the object will hit the ground approximately $14.50$ seconds after being dropped.

Choice A is incorrect. The function estimates that $7.25$ seconds after being dropped, the object's height will be $- 16 {(7.25)}^{2} + 3,364$ feet, or $2,523$ feet, above the ground.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.