Question

a helicopter flies 62 miles east, then turns and flies at an angle of 45° north of east for 199 miles. what is the direction of the helicopters resultant vector? measure the angle from the positive x - axis. |\overrightarrow{r}| = 246.77 miles \theta = ?° round your answer to the nearest hundredth.

Explanation:

Step1: Find the x - components of the two vectors

The first vector (eastward) has a magnitude of \(62\) miles and an angle of \(0^{\circ}\) from the positive x - axis. So its x - component \(x_1=62\cos(0^{\circ}) = 62\).
The second vector has a magnitude of \(199\) miles and an angle of \(45^{\circ}\) from the positive x - axis. So its x - component \(x_2 = 199\cos(45^{\circ})=199\times\frac{\sqrt{2}}{2}\approx199\times0.7071\approx140.71\).
The total x - component of the resultant vector \(x=x_1 + x_2=62 + 140.71 = 202.71\).

Step2: Find the y - components of the two vectors

The first vector (eastward) has a magnitude of \(62\) miles and an angle of \(0^{\circ}\) from the positive x - axis. So its y - component \(y_1=62\sin(0^{\circ}) = 0\).
The second vector has a magnitude of \(199\) miles and an angle of \(45^{\circ}\) from the positive x - axis. So its y - component \(y_2=199\sin(45^{\circ})=199\times\frac{\sqrt{2}}{2}\approx140.71\).
The total y - component of the resultant vector \(y=y_1 + y_2=0 + 140.71=140.71\).

Step3: Calculate the angle \(\theta\)

We know that \(\tan\theta=\frac{y}{x}\). Substituting \(x = 202.71\) and \(y = 140.71\) into the formula, we get \(\tan\theta=\frac{140.71}{202.71}\approx0.6942\).
Then \(\theta=\arctan(0.6942)\approx34.73^{\circ}\).

Answer:

\(34.73\)