Question

Three baseball players are playing catch. Alexa is directly south of Kyle and directly west of Barry. Alexa and Barry are 3 meters apart and Barry and Kyle are 6 meters apart. How far apart are Alexa and Kyle? If necessary, round to the nearest tenth.

meters

Answer 1

To determine the distance between Alexa and Kyle, we can use the information given about their positions:

1. **Positioning the Players:**
- Let's place Kyle at the origin of a coordinate system: \( K(0, 0) \).
- Since Alexa is directly south of Kyle, her position will be \( A(0, y) \) where \( y < 0 \).
- Since Alexa is directly west of Barry, and they are 3 meters apart, Barry's position will be \( B(x, y) \) where \( x > 0 \).

2. **Distance between Alexa and Barry:**
- The distance between Alexa and Barry is given as 3 meters:
\[
|x| = 3 \quad \Rightarrow \quad x = 3
\]
- Thus, Barry's coordinates become \( B(3, y) \).

3. **Distance between Barry and Kyle:**
- The distance between Barry and Kyle is given as 6 meters:
\[
\sqrt{(3 - 0)^2 + (y - 0)^2} = 6
\]
- Squaring both sides, we get:
\[
3^2 + y^2 = 6^2
\]
\[
9 + y^2 = 36
\]
\[
y^2 = 36 - 9 = 27
\]
\[
y = -\sqrt{27} = -3\sqrt{3} \quad \text{(since y is negative)}
\]

4. **Coordinates of Each Player:**
- Now we have:
- Kyle: \( K(0, 0) \)
- Alexa: \( A(0, -3\sqrt{3}) \)
- Barry: \( B(3, -3\sqrt{3}) \)

5. **Distance between Alexa and Kyle:**
- Now, we can find the distance between Alexa and Kyle:
\[
\text{Distance} = \sqrt{(0 - 0)^2 + \left(0 - (-3\sqrt{3})\right)^2}
\]
\[
= \sqrt{0 + (3\sqrt{3})^2}
\]
\[
= \sqrt{27} = 3\sqrt{3}
\]
- Approximating \( 3\sqrt{3} \):
\[
\sqrt{3} \approx 1.732 \implies 3\sqrt{3} \approx 3 \times 1.732 \approx 5.196
\]

Finally, rounding to the nearest tenth:
\[
\text{Distance between Alexa and Kyle} \approx 5.2 \text{ meters}
\]

Thus, the distance between Alexa and Kyle is **5.2 meters**.