Answer:
[tex]\text{1. }2.53^{\circ},\\\text{2. }792,772.98\:\mathrm{ft}[/tex]
Step-by-step explanation:
Start by converting miles to feet:
[tex]150\text{ miles}=150\cdot 5280\text{ feet}=792000\text{ feet}[/tex]
Form a right triangle with the plane's displacement marking the hypotenuse of this triangle. We can now use basic trig for a right triangle to solve for the angle of descent.
Let the angle of descent be [tex]\theta[/tex]. In a right triangle, the tangent of an angle is equal to its opposite side divide by its adjacent side.
Therefore, we have:
[tex]\tan \theta=\frac{35000}{792000},\\\\\theta =\arctan(\frac{35000}{792000})=2.53036411\approx \boxed{2.53^{\circ}}[/tex]
In all right triangles, [tex]a^2+b^2=c^2[/tex], where [tex]c[/tex] is the hypotenuse of the triangle and [tex]a[/tex] and [tex]b[/tex] are the two legs of the triangle (Pythagorean Theorem).
Therefore, the length of the plane's path is:
[tex]35,000^2+792,000^2=c^2,\\c^2=628489000000,\\c=\sqrt{628489000000}=792772.981376\approx \boxed{792,772.98\:\mathrm{ft}}[/tex]
