sat-dog

Choice B is correct. It's given that for the function $h$, $h\left(n\right)$ 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 $85\%$ 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 $h\left(n\right)=a{\left(\frac{p}{100}\right)}^n$, 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, $a=22$. Since the function $h$ estimates that the maximum height reached after each time the ball hits the ground is $85\%$ of the maximum height reached after the previous time the ball hit the ground, $p=85$. Substituting $22$ for $a$ and $85$ for $p$ in the equation $h\left(n\right)=a{\left(\frac{p}{100}\right)}^n$ yields $h\left(n\right)=22{\left(\frac{85}{100}\right)}^n$, or $h\left(n\right)=22{\left(0.85\right)}^n$.

Choice A is incorrect. This function estimates that the maximum height reached after each time the ball hits the ground is $22\%$, not $85\%$, 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 $22\%$, not $85\%$, 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.