Question

modeling real life two wolves spot a deer in a field. the positions of the animals are shown. which wolf is closer to the deer? chart: x-axis (distance (m)) 0–56, y-axis (distance (m)) 0–32+, wolf a at (8,16), wolf b at (16,4), deer at (48,32) options: wolf a, wolf b

Explanation:

Step1: Determine coordinates

Find coordinates of Wolf A, Wolf B, and Deer. Let's assume each grid square is 8 m (since x-axis has marks at 0,8,16,... and y-axis same).
- Wolf A: \((8, 16)\)
- Wolf B: \((16, 4)\)
- Deer: \((48, 32)\)

Step2: Calculate distance for Wolf A

Use distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
For Wolf A to Deer:
\(x_1 = 8, y_1 = 16\); \(x_2 = 48, y_2 = 32\)
\(d_A = \sqrt{(48 - 8)^2 + (32 - 16)^2} = \sqrt{40^2 + 16^2} = \sqrt{1600 + 256} = \sqrt{1856} \approx 43.08\) m

Step3: Calculate distance for Wolf B

For Wolf B to Deer:
\(x_1 = 16, y_1 = 4\); \(x_2 = 48, y_2 = 32\)
\(d_B = \sqrt{(48 - 16)^2 + (32 - 4)^2} = \sqrt{32^2 + 28^2} = \sqrt{1024 + 784} = \sqrt{1808} \approx 42.52\) m

Step4: Compare distances

\(d_B \approx 42.52 < d_A \approx 43.08\), so Wolf B is closer. Wait, no—wait, maybe grid is 8? Wait, maybe I misread. Wait, x-axis: Wolf A at x=8, Wolf B at x=16, Deer at x=48. Y-axis: Wolf A at y=16, Wolf B at y=4, Deer at y=32. Wait, maybe each grid is 8 units? Wait, 48 - 8 = 40, 48 -16=32. Y: 32-16=16, 32-4=28. Wait, but maybe the grid is 8m per square? Wait, but let's check the differences. Alternatively, maybe the horizontal and vertical distances: for Wolf A, horizontal difference: 48 - 8 = 40, vertical: 32 -16=16. For Wolf B: horizontal 48 -16=32, vertical 32 -4=28. Then, using distance formula, Wolf A: \(\sqrt{40^2 +16^2}=\sqrt{1600+256}=\sqrt{1856}\approx43.08\). Wolf B: \(\sqrt{32^2 +28^2}=\sqrt{1024+784}=\sqrt{1808}\approx42.52\). Wait, but that would mean Wolf B is closer? But maybe I made a mistake. Wait, no—wait, maybe the coordinates are (x,y) where x and y are in meters. Wait, the x-axis is Distance (m), so each grid line is, say, 8m? Wait, 0,8,16,24,32,40,48,56. So each grid square is 8m. So Wolf A is at (8,16), Wolf B at (16,4), Deer at (48,32). So horizontal distance from Wolf A to Deer: 48 -8=40m, vertical: 32-16=16m. Distance: \(\sqrt{40^2 +16^2}=\sqrt{1600+256}=\sqrt{1856}\approx43.08\)m. Wolf B: horizontal 48-16=32m, vertical 32-4=28m. Distance: \(\sqrt{32^2 +28^2}=\sqrt{1024+784}=\sqrt{1808}\approx42.52\)m. So Wolf B is closer? Wait, but maybe the problem is simpler: count the number of grid units? Wait, maybe each grid is 8m, but maybe the horizontal and vertical differences. Wait, no—wait, maybe I messed up the coordinates. Wait, Wolf A: x=8, y=16. Deer: x=48, y=32. So Δx=40, Δy=16. Wolf B: x=16, y=4. Δx=32, Δy=28. Then, distance for A: \(\sqrt{40^2 +16^2}\), for B: \(\sqrt{32^2 +28^2}\). Let's compute 40²=1600, 16²=256, sum=1856. 32²=1024, 28²=784, sum=1808. Since 1808 < 1856, so \(\sqrt{1808} < \sqrt{1856}\), so Wolf B is closer? Wait, but the options are Wolf A or B. Wait, maybe I made a mistake in coordinates. Wait, looking at the graph: Wolf A is at (8,16), Wolf B at (16,4), Deer at (48,32). Wait, maybe the y-axis is distance, so vertical distance. Wait, no, the graph is a coordinate plane with x and y as distance (m). So it's a 2D plane where both axes are distance, so the distance between two points (x1,y1) and (x2,y2) is \(\sqrt{(x2 -x1)^2 + (y2 - y1)^2}\). So calculating that, Wolf B is closer. Wait, but let me check again. 48-8=40, 32-16=16. 40²+16²=1600+256=1856. 48-16=32, 32-4=28. 32²+28²=1024+784=1808. 1808 is less than 1856, so square root of 1808 is less than square root of 1856. So Wolf B is closer? Wait, but maybe the problem is using the Manhattan distance? No, the distance formula is Euclidean. Wait, maybe the grid is 8m per square, but maybe the numbers are different. Wait, maybe I misread the coordinates. Let's check the x-axis: Wolf A is at x=8, Wolf B at x=16, Deer at x=48. Y-axis: Wolf A at y=16, Wolf B at y=4, Deer at y=32. So yes, that's correct. So Wolf B is closer. Wait, but the initial thought—maybe I made a mistake. Wait, let's compute the differences again. Wolf A to Deer: x: 48-8=40, y:32-16=16. Wolf B to Deer: x:48-16=32, y:32-4=28. Then, 40 and 16: 40 is larger than 32, but 16 is smaller than 28. But the sum of squares: 40²+16²=1600+256=1856; 32²+28²=1024+784=1808. So 1808 is less, so distance is less. So Wolf B is closer. Wait, but the options are Wolf A or B. So the answer should be Wolf B? Wait, but maybe I messed up the coordinates. Wait, maybe Wolf A is at (8,16), Deer at (48,32). So horizontal distance: 40, vertical:16. Wolf B at (16,4), Deer at (48,32): horizontal 32, vertical 28. So yes, Wolf B's distance is shorter. So the answer is Wolf B.

Answer:

Wolf B