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Advanced Math / Nonlinear functions Difficulty: Easy

A ball is dropped from an initial height of $22$ feet and bounces off the ground repeatedly. The function $h$ estimates that the maximum height reached after each time the ball hits the ground is $85 percent sign$ of the maximum height reached after the previous time the ball hit the ground. Which equation defines $h$ , where $h (n)$ is the estimated maximum height of the ball after it has hit the ground $n$ times and $n$ is a whole number greater than $1$ and less than $10$ ?

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Explanation

Choice B is correct. It's given that for the function $h$ , $h (n)$ is the estimated maximum height, in feet, of the ball after it has hit the ground $n$ times. It's also given that the function $h$ estimates that the maximum height reached after each time the ball hits the ground is $85 percent sign$ of the maximum height reached after the previous time the ball hit the ground. It follows that $h$ is a decreasing exponential function that can be written in the form $h (n) = a {(\frac{p}{100})}^{n}$ , where $a$ is the initial height, in feet, the ball was dropped from and the function estimates that the maximum height reached after each time the ball hits the ground is $p %$ of the maximum height reached after the previous time the ball hit the ground. It's given that the ball is dropped from an initial height of $22$ feet. Therefore, $a equals 22$ . Since the function $h$ estimates that the maximum height reached after each time the ball hits the ground is $85 percent sign$ of the maximum height reached after the previous time the ball hit the ground, $p equals 85$ . Substituting $22$ for $a$ and $85$ for $p$ in the equation $h (n) = a {(\frac{p}{100})}^{n}$ yields $h (n) = 22 {(\frac{85}{100})}^{n}$ , or $h left parenthesis n right parenthesis equals 22 left parenthesis 0.85 right parenthesis Superscript n$ .

Choice A is incorrect. This function estimates that the maximum height reached after each time the ball hits the ground is $22 percent sign$ , not $85 percent sign$ , of the maximum height reached after the previous time the ball hit the ground.

Choice C is incorrect. This function estimates that the ball is dropped from an initial height of $85$ feet, not $22$ feet, and that the maximum height reached after each time the ball hits the ground is $22 percent sign$ , not $85 percent sign$ , of the maximum height reached after the previous time the ball hit the ground.

Choice D is incorrect. This function estimates that the ball is dropped from an initial height of $85$ feet, not $22$ feet.