Question

Samantha and Isaac are playing racquetball. Samantha hits the ball, sending it on a trajectory modeled by y=−3∣x−4∣+20, where y is the height reached by the ball, in feet, after x milliseconds. Isaac throws his racquet toward it at a trajectory modeled by y=
3
1
​
x+4. When does his racquet hit the ball
A. after 8.4 milliseconds
B. after 1.5 milliseconds
C. after 6.8 milliseconds
D. after 3.5 milliseconds

Answer 1

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:
1. For Samantha (ball):
\[
y = -3|x - 4| + 20
\]
2. 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.