To determine the new direction of the aircraft, we need to consider both the jet's speed and the wind's speed and direction. Here’s a detailed, step-by-step solution:
1. Components of the Wind's Velocity:
The wind is blowing at 120 mph at an angle of [tex]\(30^{\circ}\)[/tex] north of east. We need to resolve this wind speed into its eastward and northward components.
- Eastward Component:
The eastward component is calculated using cosine of the angle:
[tex]\[
\text{Wind East Component} = \text{Wind Speed} \times \cos(30^{\circ}) = 120 \times \cos(30^{\circ}) = 103.92304845413264 \, \text{mph}
\][/tex]
- Northward Component:
The northward component is calculated using sine of the angle:
[tex]\[
\text{Wind North Component} = \text{Wind Speed} \times \sin(30^{\circ}) = 120 \times \sin(30^{\circ}) = 59.99999999999999 \, \text{mph}
\][/tex]
2. Resultant Eastward Speed:
The jet is initially flying at 500 mph due east. When taking into account the eastward component of the wind, the resultant eastward speed becomes:
[tex]\[
\text{Resultant East Speed} = \text{Jet Speed} + \text{Wind East Component} = 500 + 103.92304845413264 = 603.9230484541326 \, \text{mph}
\][/tex]
3. Determination of the New Direction:
To find the new direction of the aircraft, we use the arctangent function to determine the angle of deviation north of east due to the northward component of the wind:
[tex]\[
\theta = \arctan\left(\frac{\text{Wind North Component}}{\text{Resultant East Speed}}\right)
\][/tex]
Substituting the values we get:
[tex]\[
\theta = \arctan\left(\frac{59.99999999999999}{603.9230484541326}\right) = 0.09902544874225092 \, \text{radians}
\][/tex]
4. Convert Radians to Degrees:
To express this angle in degrees:
[tex]\[
\theta_{\text{degrees}} = \theta \times \left(\frac{180}{\pi}\right) = 0.09902544874225092 \times \left(\frac{180}{\pi}\right) = 5.673740277320044^{\circ}
\][/tex]
Therefore, the new direction of the aircraft is approximately [tex]\(5.67^{\circ}\)[/tex] north of east. Hence, the correct answer is:
a. [tex]\(5.67^{\circ}\,N\)[/tex] of [tex]\(E\)[/tex]
