Starting from its nest an eagle flies at constant speed for 2.0 due east then 4.2 north From there the directly to same How long is in air?

Displacement for first leg of trip,

$$x_{1}=2\text{ km; }y_1=0\text{ km}$$

Displacement for second leg of trip,

$$x_2=0\text{ km; }y_2=4.2\text{ km}$$

Adding these displacements gives the total displacement as,

$$\begin{split}\vec R &=\vec R_1+\vec R_2 \\\ &=(2\hat{i}+ 0\hat{ j})+(0\hat{i}+4.2\hat{j}) \\\ &=(2\hat{i}+ 4.2\hat{j})\text{ km} \\\ |\vec R| &=\sqrt{x^2_2+y^2_2}=\sqrt{2^2+4.2^2}\text{ km} \\\ &=\boxed{4.6 \text{ km}}\end{split}$$

To find the time the eagle is in the air, we can use the equation:

$$\text{speed}=\frac{\text{distance}}{\text{time}}$$

Since the eagle is flying at a constant speed, the average speed is given by:

$$v=\frac{\text{total distance}}{\text{total time}}$$

Solving for the total time and plugging in the average speed gives:

$$t=\frac{\text{total distance}}{\text{average speed}}=\frac{|\vec{R}|}{v}$$

Substituting the values we know, we get:

$$t=\frac{4.6\text{ km}}{1.5\text{ km/min}}=\boxed{3.1 \text{ min}}$$