While playing a game, Jordan and Kyle start at the same base then each go and hide their team’s flag. Jordan walks 32 yards due east and then turns 43° north of east and walks an a... While playing a game, Jordan and Kyle start at the same base then each go and hide their team’s flag. Jordan walks 32 yards due east and then turns 43° north of east and walks an additional 30 yards. Kyle walks 32 yards due west and then turns 38° north of west and walks another 30 yards. At this point, who is closer to the base? Read more

Steps to Solve

1. Jordan's First Move

Jordan walks 32 yards due east. He starts at the origin (0, 0) and ends at (32, 0).

2. Jordan's Second Move

Next, Jordan turns 43° north of east and walks 30 yards. To find his new position, we break it into components:

• The x-coordinate: $$ x = 32 + 30 \cdot \cos(43°) $$
• The y-coordinate: $$ y = 0 + 30 \cdot \sin(43°) $$

Calculating these:

• $$ x \approx 32 + 30 \cdot 0.7314 \approx 32 + 21.942 \approx 53.942 $$
• $$ y \approx 0 + 30 \cdot 0.6819 \approx 20.457 $$

So, Jordan’s final position is approximately $(53.942, 20.457)$.

3. Kyle's First Move

Kyle walks 32 yards due west. Starting from the origin (0, 0), he ends at (-32, 0).

4. Kyle's Second Move

Kyle then turns 38° north of west and walks 30 yards. For his new position:

• The x-coordinate: $$ x = -32 + 30 \cdot \cos(180° - 38°) $$
• The y-coordinate: $$ y = 0 + 30 \cdot \sin(180° - 38°) $$

Calculating these:

• Since $\cos(180° - 38°) = -\cos(38°)$, we have: $$ x \approx -32 - 30 \cdot 0.7880 \approx -32 - 23.640 \approx -55.640 $$
• $$ y \approx 0 + 30 \cdot 0.6157 \approx 18.471 $$

So, Kyle’s final position is approximately $(-55.640, 18.471)$.

5. Calculating Distances from the Base

Now, we calculate the distances from their respective final positions to the origin (0, 0):

• Jordan's distance: $$ d_J = \sqrt{(53.942)^2 + (20.457)^2} $$
• Kyle's distance: $$ d_K = \sqrt{(-55.640)^2 + (18.471)^2} $$

Calculating these:

• For Jordan: $$ d_J \approx \sqrt{2904.717 + 418.408} \approx \sqrt{3323.125} \approx 57.66 $$
• For Kyle: $$ d_K \approx \sqrt{3095.277 + 341.683} \approx \sqrt{3436.960} \approx 58.7 $$