Question

Samantha and Isaac are playing racquetball. Samantha hits the ball, sending it on a trajectory modeled by y=−3|x−4|+20
𝑦
=
−
3
|
𝑥
−
4
|
+
20
, where y is the height reached by the ball, in feet, after x milliseconds. In a desperate attempt to keep the ball in the air, Isaac throws his racquet toward it at a trajectory modeled by y=13x+4
𝑦
=
1
3
𝑥
+
4
. When does his racquet hit the ball?(1 point)
Responses

after 6.8 milliseconds
after 6.8 milliseconds

after 3.5 milliseconds
after 3.5 milliseconds

after 1.5 milliseconds
after 1.5 milliseconds

after 8.4 milliseconds

Answer 1

To find when Isaac's racquet hits the ball, we need to set the two equations equal to each other and solve for \(x\).

1. **Samantha's equation** is:
\[
y = -3|x - 4| + 20
\]

2. **Isaac's equation** is:
\[
y = \frac{1}{3}x + 4
\]

We set these two equations equal to each other:

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

Next, we'll solve for \(x\). First, rearrange the equation:

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

Now we will consider the cases for the absolute value.

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

In this case, \(|x - 4| = x - 4\). Substituting this into the equation gives:

\[
-3(x - 4) = \frac{1}{3}x - 16
\]
\[
-3x + 12 = \frac{1}{3}x - 16
\]

To eliminate the fraction, multiply the entire equation by 3:

\[
-9x + 36 = x - 48
\]

Now, combine like terms:

\[
36 + 48 = 9x + x
\]
\[
84 = 10x
\]
\[
x = 8.4
\]

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

In this case, \(|x - 4| = -(x - 4) = 4 - x\). Substituting this into the equation:

\[
-3(4 - x) = \frac{1}{3}x - 16
\]
\[
-12 + 3x = \frac{1}{3}x - 16
\]

Multiply the entire equation by 3 to eliminate the fraction:

\[
-36 + 9x = x - 48
\]

Combine like terms:

\[
-36 + 48 = x - 9x
\]
\[
12 = -8x
\]
\[
x = -1.5
\]

### Conclusion

Considering the two solutions \(x = 8.4\) and \(x = -1.5\), the solution we are looking for within the context of the problem (when Isaac’s racquet hits the ball) is:

\[
\boxed{8.4 \text{ milliseconds}}
\]