An airplane flies from Ft. Myers to Sarasota, a
distance of 150 miles, and then turns through an
angle of 50° and flies to Orlando, a distance of 100
miles (see the figure).
(a) How far is it from Ft. Myers to Orlando?
(b) Through what angle should the pilot turn at
Orlando to return to Ft. Myers?
To solve this problem, we can use the concept of vector addition and trigonometry.
Let's start with part (a), finding the distance from Ft. Myers to Orlando.
Step 1: Split the distances into horizontal and vertical components.
- The 150-mile distance from Ft. Myers to Sarasota can be split as: 150 miles (horizontal) and 0 miles (vertical).
- The 100-mile distance from Sarasota to Orlando can be split as: 100*cos(50°) miles (horizontal) and 100*sin(50°) miles (vertical).
Step 2: Add the horizontal and vertical components separately.
- Horizontal component: 150 miles + 100*cos(50°) miles.
- Vertical component: 0 miles + 100*sin(50°) miles.
Step 3: Calculate the total distance using the Pythagorean theorem.
- Total distance = √[(horizontal component)^2 + (vertical component)^2].
Let's calculate the distance:
Horizontal component = 150 miles + 100*cos(50°) miles ≈ 150 miles + 100*0.64279 miles ≈ 150 miles + 64.279 miles ≈ 214.279 miles.
Vertical component = 0 miles + 100*sin(50°) miles ≈ 0 miles + 100*0.76604 miles ≈ 0 miles + 76.604 miles ≈ 76.604 miles.
Total distance = √[(214.279 miles)^2 + (76.604 miles)^2] ≈ √[45930.924041 miles^2 + 5861.654816 miles^2] ≈ √[51892.578857 miles^2] ≈ 228.039 miles.
Therefore, the distance from Ft. Myers to Orlando is approximately 228.039 miles.
Now, let's move on to part (b), finding the angle the pilot should turn at Orlando to return to Ft. Myers.
Step 1: Calculate the horizontal and vertical distances from Orlando to Ft. Myers.
- Horizontal distance from Orlando to Ft. Myers: 214.279 miles.
- Vertical distance from Orlando to Ft. Myers: -76.604 miles (negative because it is in the opposite direction).
Step 2: Use trigonometry to find the angle.
- Angle = arctan(vertical distance / horizontal distance).
Let's calculate the angle:
Angle = arctan(-76.604 miles / 214.279 miles) ≈ arctan(-0.357591) ≈ -20.481°.
Therefore, the pilot should turn through an angle of approximately -20.481° at Orlando to return to Ft. Myers. Note that the negative sign indicates a clockwise turn.
To solve this problem, we can use the Law of Cosines and Law of Sines. Let's break it down step-by-step:
Step 1: Calculate the distance from Ft. Myers to Orlando.
We can use the Law of Cosines to find the distance between two points given the lengths of two sides and the angle between them.
In this case, let's consider the sides:
a = 150 miles (Ft. Myers to Sarasota)
b = 100 miles (Sarasota to Orlando)
θ = 50° (angle at Sarasota)
Using the Law of Cosines, we can calculate the distance c from Ft. Myers to Orlando:
c² = a² + b² - 2ab * cos(θ)
c² = 150² + 100² - 2 * 150 * 100 * cos(50°)
c² = 22,500 + 10,000 - 30,000 * cos(50°)
c² = 32,500 - 30,000 * cos(50°)
c = √(32,500 - 30,000 * cos(50°))
Calculating this expression gives us:
c ≈ √(32,500 - 30,000 * 0.6428)
c ≈ √(32,500 - 19,280)
c ≈ √13,220
c ≈ 115.03 miles
Therefore, the distance from Ft. Myers to Orlando is approximately 115.03 miles.
Step 2: Calculate the angle to turn at Orlando to return to Ft. Myers.
We can use the Law of Sines to find the angle opposite to the side we just found (c) in the triangle formed by Ft. Myers, Sarasota, and Orlando.
In this case, we know:
a = 150 miles (Ft. Myers to Sarasota)
b = 100 miles (Sarasota to Orlando)
c = 115.03 miles (Ft. Myers to Orlando)
We want to find angle θ.
Using the Law of Sines, we have:
sin(θ) / c = sin(50°) / a
Rearranging the equation, we get:
sin(θ) = (c * sin(50°)) / a
θ = arcsin((c * sin(50°)) / a)
Substituting the values we calculated previously, we have:
θ = arcsin((115.03 * sin(50°)) / 150)
Calculating this expression gives us:
θ ≈ arcsin(0.5744)
θ ≈ 34.61°
Therefore, the pilot should turn at Orlando through an angle of approximately 34.61° to return to Ft. Myers.
To summarize:
(a) The distance from Ft. Myers to Orlando is approximately 115.03 miles.
(b) The pilot should turn at Orlando through an angle of approximately 34.61° to return to Ft. Myers.
227.56 mi
30.00 degrees