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. In a desperate attempt to keep the ball in the air, 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 6.8 milliseconds
B. After 3.5 milliseconds
C. After 8.4 milliseconds
D. After 1.5 milliseconds

Answer 1

To determine when Isaac's racquet hits the ball, we need to find the point where the two trajectories intersect. This means we need to set the two equations equal to one another.

The trajectories given are:

1. Samantha's trajectory: \( y = -3|x - 4| + 20 \)
2. Isaac's trajectory: \( y = \frac{3}{1}x + 4 = 3x + 4 \)

Setting them equal to find the intersection:

\[
-3|x - 4| + 20 = 3x + 4
\]

We need to solve this equation for \( x \). Depending on the value of \( x \), \( |x - 4| \) can result in two cases:

**Case 1: \( x - 4 \geq 0 \) (which implies \( x \geq 4 \))**
In this case, \( |x - 4| = x - 4 \).

Substituting this into the equation gives:

\[
-3(x - 4) + 20 = 3x + 4
\]

Expanding and simplifying:

\[
-3x + 12 + 20 = 3x + 4 \\
-3x + 32 = 3x + 4 \\
32 - 4 = 3x + 3x \\
28 = 6x \\
x = \frac{28}{6} = \frac{14}{3} \approx 4.67
\]

**Case 2: \( x - 4 < 0 \) (which implies \( x < 4 \))**
In this case, \( |x - 4| = -(x - 4) = 4 - x \).

Substituting this into the equation gives:

\[
-3(4 - x) + 20 = 3x + 4
\]

Expanding and simplifying:

\[
-12 + 3x + 20 = 3x + 4 \\
3x + 8 = 3x + 4 \\
8 \neq 4 \quad \text{(No solutions in this case)}
\]

Since \( x < 4 \) yields no valid solutions, we only have the solution from Case 1:

\[
x = \frac{14}{3} \approx 4.67 \text{ milliseconds}
\]

We need to check if this is one of the provided answer choices. Since \( 4.67 \) milliseconds does not match any of the given options (A. 6.8, B. 3.5, C. 8.4, D. 1.5), we will also check for other solutions or verify the calculations to see if we missed anything.

Since the problem specific is asking when Isaac's racquet hits the ball through two trajectories, it appears that the solution provided falls outside the direct answers. Thus, it’s suggesting perhaps a reevaluation of the attempt.

However, looking again and confirming, with \( \approx 4.67 \) being the only intersection, it highlights that Isaac’s racquet did not hit the ball at the tied values.

Rethinking or reviewing gaps in the problem setup could assist further with comparative measures if intended.

Conclusively with the evaluations so far, the option nearest available checks with several calculative reviews showcasing numeric interpretations.

Ultimately, the closest approximate answer might lean operational discussions and resolutions for explorations outright or based interpretations.

The trajectories intersect at approximately 4.67 milliseconds which does not allow a defined validation toward stakeholder choices provided, rather explicable from sequence logic indicating a longer journey for a conclusive match.

Lastly, there may be reflection on content clarifications intended to reformat or precision stated values.