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

Given:

Let x=34 miles

y=44 miles

`(dy)/(dt)=41 miles/hr`

To find:

`(d\theta)/(dt)`

Solution:

`Hypotenuse=√((base)^2+(Perpendicular\;side)^2)`

Using the formula

`Hypotenuse=√((34)^2+(44)^2)=55.6 miles`

`sec\theta=(Hypotenuse)/(Base)`

Using the formula

`sec\theta=(55.6)/(34)`

`tan\theta=(Perpendicular\;side)/(Base)`

Using the formula

`tan\theta=(y)/(34)`

Differentiate w.r.t t

`sec^2\theta(d\theta)/(dt)=(1)/(34)(dy)/(dt)`

Using the formula

`(d(tanx))/(dx)=sec^2 x`

Substitute the values

`((55.6)/(34))^2* (d\theta)/(dt)==(1)/(34)* 41`

`(d\theta)/(dt)=(41* (34)^2)/(34* (55.6)^2)`

`(d\theta)/(dt)=0.45 rad/hour`