Sketch a diagram describing the given situation, and then use the Distance Formula to evaluate the resultant speed.
≈ 354.3 miles per hour at an angle of 8.9^(∘) east of north
Practice makes perfect
We are given that a plane is traveling due north at a speed of 350 miles per hour and the wind is blowing from the west at a speed of 55 miles per hour. We need to evaluate the resultant speed and direction of this plane. Let's begin with sketching a diagram describing the given situation.
As we can see, the component form of the vector representing the plane velocity is ⟨ 0, 350⟩, and the component form of the vector representing the velocity of the wind is ⟨ 55,0 ⟩. Therefore, the resultant vector is the sum of these two vectors.
⟨ 0, 350⟩ + ⟨ 55,0 ⟩ = ⟨ 55, 350⟩
The vector ⟨ 55,350 ⟩ represents the resultant velocity of the plane, which we will call r.
The resultant speed of the plane is the magnitude of r. To evaluate this speed, we will use the Distance Formula. We will substitute ( , ) for the initial point and ( 55, 350) for the terminal point.
The resultant speed of the plane is approximately 354.3 miles per hour. To evaluate the resultant direction, we need to find the measure of an angle that vector r forms with a north-south line. We will call this angle θ.
To find the measure of θ, we can use one of the trigonometric ratios as the vectors form a right triangle. Let's recall that in a right triangle the tangent of an angle is a ratio between the leg opposite to this angle and leg adjacent to this angle. Using this definition, we can write an equation.
tan θ=55/350
Next we can rewrite the equation using the inverse tangent to evaluate the measure of θ.
tan θ=55/350 ⇓
θ=tan ^(-1)55/350 ≈ 8.9^(∘)
The resultant direction is approximately 8.9^(∘) east of north.
