A track in a national park has four legs from its starting point to its return. The first leg travels due east. The second, third and fourth legs follow paths southwest, due north and finally due east back to the starting point. If the distances of the first two legs are 1.5 km and 3.5 km respectively, what distances are the third and fourth legs, correct to 2 decimal places?
To solve this problem, we can break down the directions and distances given:
1. The first leg travels due east.
2. The second leg follows a southwest direction.
3. The third leg travels due north.
4. The fourth leg travels due east back to the starting point.
Let's calculate the distances for the third and fourth legs:
1. The first leg is given as 1.5 km, and it travels due east.
2. The second leg is given as 3.5 km and follows a southwest direction. Since the direction is southwest, we can split it into its east and south components using the trigonometric ratios. Let's say the distance in the east direction is x km, and in the south direction is y km. Then, according to trigonometry, we have:
x = 3.5 * cos(45°) (cosine of 45° is 1/sqrt(2) ≈ 0.707)
y = 3.5 * sin(45°) (sine of 45° is also 1/sqrt(2) ≈ 0.707)
Therefore, x ≈ 2.47 km and y ≈ 2.47 km.
3. The third leg travels due north, which means it will have the same value as the south component of the second leg. Therefore, the distance for the third leg is approximately 2.47 km.
4. The fourth leg travels due east back to the starting point. Since it is the opposite of the first leg, it will have the same distance but in the opposite direction. Therefore, the distance for the fourth leg is approximately -1.5 km.
So, the distances for the third and fourth legs are approximately 2.47 km and -1.5 km, respectively.
To solve this problem, we can start by sketching a diagram to visualize the track. Let's assume the starting point is at the origin (0,0) on a coordinate plane.
First, we will travel due east for the first leg. Since there is no change in the north-south direction, the coordinates after the first leg will be (1.5, 0).
Next, we'll travel southwest for the second leg. The distance traveled in the east direction is equal to the distance traveled in the south direction. So, the coordinates after the second leg will be (1.5 - 3.5, -3.5) or (-2, -3.5).
For the third leg, we'll travel due north. Since there is no change in the east-west direction, the coordinates after the third leg will be (-2, -3.5 + x), where x is the distance traveled in the north direction.
Lastly, we'll travel due east back to the starting point. This means we need to cancel out the southward distance we traveled in the second leg. We already know that the eastward distance is 1.5 - 3.5 = -2. So, the coordinates after the fourth leg will be (-2 + y, -3.5 + x), where y is the distance traveled in the east direction in the fourth leg.
To reach the starting point, the coordinates after the fourth leg must be (0, 0). Equating the coordinates, we get:
-2 + y = 0
-3.5 + x = 0
From the first equation, we find y = 2. From the second equation, we find x = 3.5.
Therefore, the third leg is 2 km and the fourth leg is 3.5 km.
leg2 goes 3.5/√2 south and 3.5/√2 west
So leg 3 goes 3.5/√2 north
leg4 goes 3.5/√2 - 1.5 east