To find out when Isaac's racquet hits the ball, we need to set the equations for their trajectories equal to each other and solve for \(x\):
The equations are:
- For Samantha (ball): \[ y = -3|x - 4| + 20 \]
- For Isaac (racquet): \[ y = \frac{3}{1}x + 4 = 3x + 4 \]
Setting the two equations equal to each other: \[ -3|x - 4| + 20 = 3x + 4 \]
Now, we need to consider two cases for the absolute value expression \( |x - 4| \).
Case 1: \( x - 4 \geq 0 \) (i.e., \( x \geq 4 \))
In this case, \( |x - 4| = x - 4 \). Therefore, the equation becomes: \[ -3(x - 4) + 20 = 3x + 4 \] Simplifying this: \[ -3x + 12 + 20 = 3x + 4 \] \[ -3x + 32 = 3x + 4 \] Adding \(3x\) to both sides: \[ 32 = 6x + 4 \] Subtracting \(4\) from both sides: \[ 28 = 6x \] Dividing both sides by \(6\): \[ x = \frac{28}{6} = \frac{14}{3} \approx 4.67 \]
Case 2: \( x - 4 < 0 \) (i.e., \( x < 4 \))
In this case, \( |x - 4| = -(x - 4) = 4 - x \). Therefore, the equation becomes: \[ -3(4 - x) + 20 = 3x + 4 \] Simplifying this: \[ -12 + 3x + 20 = 3x + 4 \] \[ 3x + 8 = 3x + 4 \] This simplifies to: \[ 8 = 4 \] This case has no valid solutions since it leads to a contradiction.
Conclusion
The only valid solution occurs when \(x = \frac{14}{3} \approx 4.67\), which does not directly match the choices provided. However, we will check which answer is the closest.
Calculating the values for the options, we have:
- A: 8.4
- B: 1.5
- C: 6.8
- D: 3.5
Among the choices, the closest one that makes sense is C: after 6.8 milliseconds, as it is the closest to our calculation \(4.67\).
Final Answer:
C. after 6.8 milliseconds.