In exact fractional form

Question

A boat starts off 34 miles directly west from the city of Uniontown. It travels due north at a speed of 41 miles per hour. After travelling 44 miles, how fast (in radians per hour) is the angle opposite the northward path changing?

Answer 1

Let x = distance travelled by the boat in t hours

Given: dx/dt = 41.

Draw a right triangle with horizontal leg 34 and vertical leg x; Let θ be the angle opposite the side labelled x.

Find dθ/dt when x = 44:

tanθ = x / 34

sec²θ (dθ/dt) = (1/34)(dx/dt) = 41/34

So, dθ/dt = (41/34)cos²θ = (41/34)[34²/ (34²+44²)] = 1394/3092 = 697/1546 radians/hour

Answer 2

How fast is the angle θ changing?

Use: tanθ = opposite/adjacent = y/x

x is constant (-34 miles), so it can be plugged in later

Isolate θ to one side:

θ = arctan(y/x) = arctan(-y/34)

dθ/dt = d(arctan(-y/34))/dt = d(arctan(-y/34))/dy*dy/dt

differentiate the right side with respect to y, plug in 41 for dy/dt and 44 for y to get dθ/dt

I encourage you to do the math yourself! Hope the visualization helps