A commuter plane flies from City A to City B, a distance of 90 mi due
north. Due to bad weather, the plane is redirected at take-off to a
heading N 60° W (60° west of north). After flying 57 mi, the plane is
directed to turn northeast and fly directly toward City B. To the
nearest tenth, how many miles did the plane fly on the last leg of
the trip?
1. Based on the diagram provided for this problem, which measures of the triangle do you know?
2. What are the values of these measures?
3. Describe the part of the triangle you need to find.
4. What concept will you use to write an equation? What is the equation?
5. Solve the equation. What is the distance of the last leg of the trip?
surely you can answer some of this if you draw a diagram. Think about the law of cosines for part 5.
1. Based on the information provided, we know the following measures of the triangle:
- The distance flown from City A to the redirected heading is 57 miles.
- The angle between the redirected heading and the heading towards City B is 60°.
2. The values of these measures are:
- Distance flown from City A to redirected heading: 57 miles.
- Angle between the redirected heading and the heading towards City B: 60°.
3. The part of the triangle we need to find is the distance of the last leg of the trip, which is the distance from the redirected heading to City B.
4. To find the distance of the last leg of the trip, we can use the concept of trigonometry and the Law of Cosines:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, a is the distance flown from City A to the redirected heading (57 miles), b is the distance from the redirected heading to City B (which we want to find), and C is the angle between the two sides a and b (60°).
5. Plugging in the values into the equation, we have:
b^2 = 57^2 + (90 - b)^2 - 2 * 57 * (90 - b) * cos(60°)
Solving this equation will give us the distance of the last leg of the trip.
1. Based on the problem description, we know the following measures of the triangle:
- The distance from City A to City B, which is 90 miles.
- The distance traveled on the first leg, which is 57 miles.
2. The values of these measures are:
- Distance from City A to City B: 90 miles
- Distance traveled on the first leg: 57 miles
3. We need to find the distance of the last leg of the trip.
4. To find the distance of the last leg of the trip, we can use the concept of the Pythagorean Theorem. The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
We can write the equation as follows:
(distance of the last leg)^2 = (distance from City A to City B)^2 - (distance traveled on the first leg)^2
5. Now, let's solve the equation using the given values:
(distance of the last leg)^2 = (90 miles)^2 - (57 miles)^2
(distance of the last leg)^2 = 8100 miles^2 - 3249 miles^2
(distance of the last leg)^2 = 4851 miles^2
Taking the square root of both sides, we get:
distance of the last leg ≈ √4851 miles
Rounding to the nearest tenth, the distance of the last leg of the trip is approximately 69.7 miles.