Rocio drops a ball from a height of $4$ meters. Rocio observes that each time the ball bounces, it reaches a peak height which is $79 \%$ of the previous peak height, as shown. Which of the following equations correctly models the ball's peak height, $h$ , in meters, after $b$ bounces? $\newline$ Choose $1$ answer: $\newline$ (A) $h=4 \cdot 0.79^{(b-1)}$ $\newline$ (B) $h=4 \cdot 0.79^{b}$ $\newline$ (C) $h=4-0.79 \cdot b^{2}$ $\newline$ (D) $h=4 \cdot\left(1-0.79 \cdot b^{2}\right)$

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Algebra 1

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Q. Rocio drops a ball from a height of

4

meters. Rocio observes that each time the ball bounces, it reaches a peak height which is

79 \%

of the previous peak height, as shown. Which of the following equations correctly models the ball's peak height,

h

, in meters, after

b

bounces?

\newline

Choose

1

answer:

\newline

(A)

h=4 \cdot 0.79^{(b-1)}

\newline

(B)

h=4 \cdot 0.79^{b}

\newline

(C)

h=4-0.79 \cdot b^{2}

\newline

(D)

h=4 \cdot\left(1-0.79 \cdot b^{2}\right)

Understand the problem: Understand the problem. $\newline$ We need to find an equation that models the peak height of the ball after it bounces. The initial height is $4$ meters, and after each bounce, the ball reaches $79\%$ of its previous height.
Analyze the answer choices: Analyze the answer choices. $\newline$ We are given four different equations to choose from. We need to determine which equation correctly represents the situation where the height of the ball after each bounce is $79\%$ of the height of the previous bounce.
Evaluate each option: Evaluate each option. $\newline$ (A) $h=4\times0.79^{(b-1)}$ suggests that the initial bounce is not counted, which is incorrect because the initial height should be considered as the first peak. $\newline$ (B) $h=4\times0.79^{b}$ suggests that the height after $b$ bounces is $4$ meters multiplied by $0.79$ raised to the power of $b$ , which correctly accounts for the initial height and the reduction after each bounce. $\newline$ (C) $h=4-0.79\times b^{2}$ suggests a linear and quadratic decrease, which does not match the description of the problem. $\newline$ (D) $h=4\times(1-0.79\times b^{2})$ also suggests a quadratic decrease, which is incorrect for the same reason as option C.
Choose the correct equation: Choose the correct equation. $\newline$ Based on the description of the problem, the height of the ball after each bounce is a fixed percentage ( $79\%$ ) of the height of the previous bounce. This is an exponential decay situation, not linear or quadratic. Therefore, the correct equation must involve an exponent. The only option that correctly represents an exponential decay where the initial height is included in the first peak is option (B).