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= 1/3x + 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 find the value of \( x \) where the two trajectories are equal:

1. The trajectory of the ball is given by:
\[
y = -3|x - 4| + 20
\]

2. The trajectory of Isaac's racquet is given by:
\[
y = \frac{1}{3}x + 4
\]

We can set these two equations equal to each other to find the intersection points:

\[
-3|x - 4| + 20 = \frac{1}{3}x + 4
\]

Now, we'll solve this equation. First, let's isolate the absolute value term:

\[
-3|x - 4| = \frac{1}{3}x + 4 - 20
\]
\[
-3|x - 4| = \frac{1}{3}x - 16
\]
\[
|x - 4| = -\frac{1}{9}x + \frac{16}{3}
\]

Now, we will consider two cases for the absolute value \(|x - 4|\).

### Case 1: \(x - 4 \geq 0\) (i.e., \(x \geq 4\))

In this case, \(|x - 4| = x - 4\):
\[
x - 4 = -\frac{1}{9}x + \frac{16}{3}
\]
Multiply both sides by 9 to eliminate the fraction:
\[
9(x - 4) = -x + 48
\]
\[
9x - 36 = -x + 48
\]
\[
10x = 84
\]
\[
x = 8.4
\]

### Case 2: \(x - 4 < 0\) (i.e., \(x < 4\))

In this case, \(|x - 4| = -(x - 4) = 4 - x\):
\[
4 - x = -\frac{1}{9}x + \frac{16}{3}
\]
Multiply both sides by 9:
\[
9(4 - x) = -x + 48
\]
\[
36 - 9x = -x + 48
\]
\[
36 + 8x = 48
\]
\[
8x = 12
\]
\[
x = 1.5
\]

Now we have two potential solutions: \(x = 8.4\) and \(x = 1.5\).

For the question, we need to identify which options are valid:

- **A. after 8.4 milliseconds**
- **B. after 1.5 milliseconds**
- **C. after 6.8 milliseconds**
- **D. after 3.5 milliseconds**

The valid options for when Isaac's racquet hits the ball are A and B.

### Conclusion

Since the initial question considers the point of intersection, both 1.5 ms and 8.4 ms are valid points of intersection, but since the options only list one of these as acceptable based on your problem format, we can conclude that Isaac’s racquet hits the ball after:

**B. after 1.5 milliseconds.**