$$ t \geq 0 $$ 6 Milliple Cholce 10 points Write an equation that models the following siluation Samantha's hair was known to grow very rapidly il began at a length ef 6 in and grow at a rate of 145 a wock $$ \begin{array}{l} y=6(1.14)^{z} \ y=6(1+14)^{2} \ y=6(0.14)^{x} \ y=6(0.86)^{2} \end{array} $$

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To model Samantha's hair growth, we need to establish an exponential growth equation based on the given information:

1. **Initial Length**: Samantha's hair starts at 6 inches.
2. **Growth Rate**: Her hair grows at a rate of 14.5% per week (assuming "145 a wock" was a typographical error and should be "14.5% per week").

The general form of an exponential growth equation is:

$$
y = y_0 (1 + r)^t
$$

Where:
- $$ y $$ = future length of the hair
- $$ y_0 $$ = initial length (6 inches)
- $$ r $$ = growth rate per time period (14.5% or 0.145)
- $$ t $$ = number of time periods (weeks)

Plugging in the values:

$$
y = 6(1 + 0.145)^t = 6(1.145)^t
$$

Among the provided options, the closest correct model is:

$$
y = 6(1.14)^z
$$

This equation represents the hair growing from an initial length of 6 inches at an approximate growth rate of 14% per time period $$ z $$.

**Correct Equation:**

$$
y = 6(1.14)^{z}
$$
The correct equation modeling Samantha's hair growth is: $$ y = 6(1.14)^{z} $$

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