Three ballet dancers are position on stage. If Tina is 7 feet straight behind Maria and 6 feet directly left of Shawn, how far is Maria from Shawn? If necessary, round to the nearest tenth.
The Correct Answer and Explanation is:
To solve this problem, we need to determine the distance between Maria and Shawn based on their positions relative to Tina.
Step 1: Visualizing the setup
We can assume a coordinate system where we place the dancers on a flat plane. Tina is described as being 7 feet directly behind Maria and 6 feet directly left of Shawn. We can think of the positions of the dancers as forming a right triangle.
Let’s place Tina at the origin of the coordinate plane, or at point ((0, 0)). From the description:
- Tina is 7 feet directly behind Maria. In a coordinate plane, “behind” would correspond to moving along the negative y-axis. Therefore, Maria is at ((0, 7)) because she is 7 feet in front of Tina (in the positive y-direction).
- Tina is 6 feet directly left of Shawn. Being “left” corresponds to moving along the negative x-axis, so Shawn is at ((-6, 0)) since he is 6 feet to the left of Tina.
Step 2: Applying the distance formula
Now, we need to calculate the distance between Maria and Shawn. The distance formula in a 2D plane is given by:
[
d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}
]
Where ((x_1, y_1)) is Maria’s position, and ((x_2, y_2)) is Shawn’s position. From the coordinates:
- Maria: ((0, 7))
- Shawn: ((-6, 0))
Substitute these into the formula:
[
d = \sqrt{(-6 – 0)^2 + (0 – 7)^2}
]
[
d = \sqrt{(-6)^2 + (-7)^2}
]
[
d = \sqrt{36 + 49}
]
[
d = \sqrt{85}
]
[
d \approx 9.2 \, \text{feet}
]
Step 3: Conclusion
The distance between Maria and Shawn is approximately 9.2 feet.
This distance is derived using basic geometry and the distance formula, which calculates the straight-line distance between two points on a coordinate plane. By organizing the positions of the dancers and applying the formula correctly, we arrive at the solution.