HPC 1.6, 5.1-5.2 practice 2

(Calculator) The shortest side of a triangle with angles [math]40^\circ, 60^\circ[/math], and [math]80^\circ[/math] has a length of [math]9[/math]. What is the length of the longest side?

(Calculator) A tower is 100 feet tall. At a particular time of day, the tower cast a 150-foot-long shadow. What is the angle of elevation from the end of the shadow to the top of the tower?

(Calculator) Determine the values of [math]b[/math] that will produce 2 triangles if [math]a=20[/math] and [math]\angle B=40^\circ[/math].

(Calculator) Consider triangle [math]ABC[/math] with [math]\angle A=0.5, AB=9[/math], and [math]BC=6[/math]. Find [math]AC[/math]. If you have two answers, write the biggest one.

Find the vector with a magnitude of 4 pointing in the same direction as [math]\overrightarrow{u}=\langle 7,24\rangle[/math].

The force vectors [math]\overrightarrow{F_1}=\langle19,10\rangle[/math] and [math]\overrightarrow{F_2}=\langle5,17\rangle[/math] are simultaneously acting on a point [math]P[/math]. Find a third vector [math]\overrightarrow{F_3}[/math] so that equilibrium takes place.

(Calculator) Suppose you are flying an airplane in a direction N[math]20^\circ[/math]E at 400 km/hr. A wind blowing in the direction N[math]65^\circ[/math]E at 60 km/hr. Let [math]\vec{v}[/math] be the velocity vector of the plane and [math]\vec{w}[/math] be the velocity vector of the wind.

(a) Find [math]\vec{v}[/math]

(b) Find [math]\vec{w}[/math]

(c) Find the magnitude and direction of the resulting velocity vector

(d) Use the law of sine or cosine to determine the magnitude of the resulting velocity vector.

(Calculator) A force of 60 pounds on a rope is used to pull a box up a ramp inclined at [math]12^\circ[/math] from the horizontal. The figure shows that the rope forms an angle of [math]38^\circ[/math] with the horizontal. How much work is done pulling the box 20 feet along the ramp? Round to the nearest integer.

(Calculator) Given the vector [math]\vec{u}=3\textbf{i}+5\textbf{j}[/math] and [math]\vec{v}=-6\textbf{i}+\textbf{j}[/math], answer the following:

(a) Find [math]-2\vec{u}+4\vec{v}[/math].

(b) If a vector in the same direction as [math]\vec{v}[/math] whose magnitude is 5, which of the following is the vector?
(A) [math]\vec{u}=\dfrac{1}{\sqrt{8}}(3\textbf{i}+5\textbf{j})[/math]
(B) [math]\vec{u}=\dfrac{1}{\sqrt{8}}(15\textbf{i}+25\textbf{j})[/math]
(C) [math]\vec{u}=\dfrac{1}{\sqrt{34}}(15\textbf{i}+25\textbf{j})[/math]
(D) [math]\vec{u}=\dfrac{1}{\sqrt{34}}(3\textbf{i}+5\textbf{j})[/math]
(E) [math]\vec{u}=15\textbf{i}+25\textbf{j})[/math]

(c) [math]\vec{u}[/math] and [math]\vec{v}[/math] are
(A) parallel
(B) orthogonal
(C) neither

(d) Find the projection of [math]\vec{u}[/math] onto [math]\vec{v}[/math]. Leave your answer in exact form (don’t round).