To find when Isaac's racquet hits the ball, we need to set their height equations equal to each other, since they will hit at the same height at the same time.
The trajectories are modeled by the following equations:
- Samantha's ball: \( y = -3|x - 4| + 20 \)
- Isaac's racquet: \( y = \frac{3}{1}x + 4 \) which simplifies to \( y = 3x + 4 \)
Now we can set the two equations equal to find the \(x\) values:
\[ -3|x - 4| + 20 = 3x + 4 \]
To solve this equation, we need to consider two cases for the absolute value.
Case 1: \(x - 4 \geq 0\) (which implies \(x \geq 4\))
In this case, \(|x - 4| = x - 4\). Substitute this into the equation:
\[ -3(x - 4) + 20 = 3x + 4 \]
Simplifying the left side:
\[ -3x + 12 + 20 = 3x + 4 \]
Combine like terms:
\[ -3x + 32 = 3x + 4 \]
Adding \(3x\) to both sides:
\[ 32 = 6x + 4 \]
Subtracting \(4\) from both sides:
\[ 28 = 6x \]
Dividing by \(6\):
\[ x = \frac{28}{6} = \frac{14}{3} \approx 4.67 \text{ milliseconds} \]
Case 2: \(x - 4 < 0\) (which implies \(x < 4\))
In this case, \(|x - 4| = -(x - 4) = 4 - x\). Substitute this into the equation:
\[ -3(4 - x) + 20 = 3x + 4 \]
Simplifying the left side:
\[ -12 + 3x + 20 = 3x + 4 \]
Combine like terms:
\[ 8 + 3x = 3x + 4 \]
Subtracting \(3x\) from both sides:
\[ 8 = 4 \]
This is not possible, which means there are no solutions in this case.
Conclusion
The only solution occurs in Case 1 where \(x \approx 4.67\) milliseconds. This time does not match any of the options provided.
Since the given options must contain a typo due to the calculations, none of the choices listed (A, B, C, D) directly match our solution.
However, if we reconsider the range for which we can choose (between 1.5, 3.5, 6.8, and 8.4), it appears that 6.8 milliseconds would be the closest estimate.
Therefore, if forced to choose among the given options, we might select:
C. after 6.8 milliseconds, but please note this does not perfectly align with our computed value of approximately \(4.67\) milliseconds.