To find the equation that models the height of the basketball, [tex]\( H \)[/tex], after [tex]\( b \)[/tex] bounces given that the height decreases exponentially by 27.8% each time, we start by understanding the relationship between the initial height and the percentage decrease.
1. Initial Height: The basketball is dropped from an initial height of 6 feet.
2. Percentage Decrease: The height of the basketball bounce decreases by 27.8% after each bounce.
To determine the height after each bounce, first, consider the remaining percentage of the height after each bounce. This can be calculated as:
[tex]\[ \text{Remaining Percentage} = 100\% - 27.8\% = 72.2\% \][/tex]
Next, convert this percentage to a decimal for calculation:
[tex]\[ \text{Remaining Percentage in Decimal} = \frac{72.2}{100} = 0.722 \][/tex]
Now, we can model the height after [tex]\( b \)[/tex] bounces with an exponential decay formula. If the initial height is 6 feet, and with each bounce the height is multiplied by 0.722, the height [tex]\( H \)[/tex] after [tex]\( b \)[/tex] bounces is given by:
[tex]\[ H = 6 \times (0.722)^b \][/tex]
This equation demonstrates that after each bounce, the height is 72.2% of the height of the previous bounce.
Therefore, among the given options, the correct model for the height of the basketball after [tex]\( b \)[/tex] bounces is:
[tex]\[ \text{Option A: } H = 6(0.722)^b \][/tex]
