Question

An airplane travels at a speed of 400 miles/hour at an angle 30° north of west. The wind is blowing 60 miles/hour at an angle 30° east of north. Use this information to answer the questions that follow.

Part A
Write the plane’s vector, p, in component form. Round your answer to the nearest hundredth.
A. 400i - 60j
B. 400i + 60j
C. 400i + 60sin(30°)j
D. 400cos(30°)i - 60sin(30°)j

Answer 1

The plane's vector p in component form is -346.41i + 200j when resolved into horizontal and vertical components using trigonometry.

Step-by-step explanation:

The plane's velocity vector, p, can be represented in component form using trigonometric functions to resolve the vector into its horizontal (i-component) and vertical (j-component) coordinates. The plane's speed is 400 miles per hour at an angle of 30° north of west.

To find the horizontal (x) component:
400 cos(30°) = 400 × (√3/2) = 346.41

To find the vertical (y) component:
400 sin(30°) = 400 × (1/2) = 200.00

Therefore, the plane's vector p in component form is 346.41i + 200.00j, which rounds to 346.41i + 200j when rounded to the nearest hundredth. However, since westward and northward are positive directions in this context, the westward (horizontal) component should be negative, and the northward (vertical) component positive. Therefore, the correct answer in component form is -346.41i + 200j.