Three deer, A, B, and C, are grazing in a field. Deer B is located 63.4 m from deer A at an angle of 50.8 ° north of west. Deer C is located 76.7 ° north of east relative to deer A. The distance between deer B and C is 96.7 m. What is the distance between deer A and C?
To find the distance between deer A and C, we need to break down the problem into components and use trigonometry.
1. Let's find the x and y components of the position of deer B relative to deer A.
- The x-component of B with respect to A is given by Bx = r * cos(θ), where r is the distance between A and B and θ is the angle north of west.
- Plugging in the values, Bx = 63.4 m * cos(50.8 °).
- Calculating Bx, we get Bx ≈ 40.25 m.
- The y-component of B with respect to A is given by By = r * sin(θ), where r is the distance between A and B and θ is the angle north of west.
- Plugging in the values, By = 63.4 m * sin(50.8 °).
- Calculating By, we get By ≈ 48.35 m.
2. Now, let's find the x and y components of the position of deer C relative to deer A.
- The x-component of C with respect to A is given by Cx = r * cos(θ), where r is the distance between A and C and θ is the angle north of east.
- Plugging in the values, Cx = 76.7 m * cos(76.7 °).
- Calculating Cx, we get Cx ≈ 76.7 m * -0.288.
- Therefore, Cx ≈ -22.13 m (negative because it is west of A).
- The y-component of C with respect to A is given by Cy = r * sin(θ), where r is the distance between A and C and θ is the angle north of east.
- Plugging in the values, Cy = 76.7 m * sin(76.7 °).
- Calculating Cy, we get Cy ≈ 76.7 m * 0.958.
- Therefore, Cy ≈ 73.63 m.
3. Now, let's find the x and y components of the position of deer C relative to deer B.
- The x-component of C with respect to B is given by Cx = r * cos(θ), where r is the distance between B and C and θ is the angle between BC and the positive x-axis.
- Plugging in the values, Cx = 96.7 m * cos(180 °).
- Calculating Cx, we get Cx ≈ -96.7 m (since C is to the west of B).
- The y-component of C with respect to B is given by By = r * sin(θ), where r is the distance between B and C and θ is the angle between BC and the positive x-axis.
- Plugging in the values, By = 96.7 m * sin(180 °).
- Calculating By, we get By ≈ 0 m.
4. Finally, we can find the total displacement between deer A and C by adding up the x and y components.
- The x-component sum, Sx = Bx + Cx = 40.25 m + (-22.13 m) = 18.12 m.
- The y-component sum, Sy = By + Cy = 0 m + 73.63 m = 73.63 m.
5. To find the distance between A and C, we use the Pythagorean theorem.
- AC^2 = Sx^2 + Sy^2.
- AC^2 = (18.12 m)^2 + (73.63 m)^2.
- Calculating AC, we get AC ≈ √(329.18 m^2 + 5408.99 m^2).
- Therefore, AC ≈ √(5738.17 m^2).
- AC ≈ 75.74 m.
Therefore, the distance between deer A and C is approximately 75.74 m.
To find the distance between deer A and C, we first need to find the coordinates of deer B and C relative to A.
Let's set up a coordinate system with deer A as the origin.
Step 1: Calculate the x-coordinate of deer B relative to A
We know that the distance between A and B is 63.4 m, and the angle is 50.8° north of west.
To find the x-coordinate of B relative to A, we use the cosine function because the angle is west of north.
cos(angle) = adjacent/hypotenuse
cos(50.8°) = x-coordinate/63.4 m
Rearranging the equation:
x-coordinate = cos(50.8°) * 63.4 m
Calculating the x-coordinate:
x-coordinate = cos(50.8°) * 63.4 m = 40.88 m
Step 2: Calculate the y-coordinate of deer B relative to A
We know that the distance between A and B is 63.4 m, and the angle is 50.8° north of west.
To find the y-coordinate of B relative to A, we use the sine function because the angle is west of north.
sin(angle) = opposite/hypotenuse
sin(50.8°) = y-coordinate/63.4 m
Rearranging the equation:
y-coordinate = sin(50.8°) * 63.4 m
Calculating the y-coordinate:
y-coordinate = sin(50.8°) * 63.4 m = 48.75 m
So the coordinates of deer B relative to A are (40.88 m, 48.75 m).
Step 3: Calculate the coordinates of deer C relative to A
We know that the angle between A and C is 76.7° north of east, and the distance between B and C is 96.7 m.
To find the coordinates of C relative to A, we need to add the x-coordinate of B to the distance between B and C, and add the y-coordinate of B to the product of the sine of the angle and the distance between B and C.
x-coordinate of C = x-coordinate of B + distance between B and C
x-coordinate of C = 40.88 m + 96.7 m = 137.58 m
y-coordinate of C = y-coordinate of B + (distance between B and C) * sin(angle between A and C)
y-coordinate of C = 48.75 m + (96.7 m) * sin(76.7°)
y-coordinate of C = 48.75 m + (96.7 m) * 0.976
Calculating the y-coordinate:
y-coordinate of C = 48.75 m + (96.7 m) * 0.976 = 141.06 m
So the coordinates of deer C relative to A are (137.58 m, 141.06 m).
Step 4: Calculate the distance between A and C
To find the distance between A and C, we use the distance formula.
distance = √((x₂ - x₁)² + (y₂ - y₁)²)
Plugging in the values:
distance = √((137.58 m - 0 m)² + (141.06 m - 0 m)²)
Calculating the distance:
distance = √(137.58 m)² + (141.06 m)² = √(18906.4764 m² + 19893.3636 m²) = √38800.84 m²
Finally, simplifying the answer:
distance = 196.96 m
Therefore, the distance between deer A and C is approximately 196.96 m.