An airplane flies at an altitude of y = 5 miles toward a point directly over an observer. The speed of the plane is 400 miles per hour. Find the rates at which the angle of elevation θ is...

An airplane flies at an altitude of

y = 5

miles toward a point directly over an observer. The speed of the plane is 400 miles per hour. Find the rates at which the angle of elevation θ is changing when the angle is

θ = 30°,
θ = 60°,

and

θ = 80°.

Asked on by jcamacho55

1 Answer | Add Yours

Top Answer

lemjay's profile pic

lemjay | High School Teacher | (Level 3) Senior Educator

Posted on

The attachment is the figure that represents the given in our problem.

To solve for the rate of change of angle of elevation, let's apply the formula of tangent function.

`tan theta=(o p p o s i te) / (a d ja cent)`

For our triangle, the tangent is

`tan theta =5/x`

Then, isolate the x.

`x=5/tan theta `          (Let this be EQ1).

Next, lets take the derivative of our tangent with respect to t.

`d/dt (tan theta)=d/dt (5/x)`

`sec^2 theta (d theta)/dt=-5/x^2 dx/dt`

Take note that dx/dt is the rate of change of horizontal line in our triangle. The rate in which it is changing is the same as with the speed of the plane. However, the horizontal line becomes smaller as the plane approaches the observer. So the value of dx/dt is -400.

`sec^2theta (d theta)/dt =-5/x^2(-400)`

`sec^2theta (d theta)/dt =2000/x^2`

Then, isolate d theta/dt.

`(d theta)/dt=2000/(x^2sec^2theta) `

`(d theta)/dt=(2000cos^2theta)/x^2`      (Let this be EQ2.)

Use EQ1 and EQ2 to solve for the rate of change of elevation given the specified angles.

(a) `theta =30^o`

Plug-in the value of theta to EQ1 to get the value of x.

`x=5/tan (30^o)=5/(sqrt3/3)=5*3/sqrt3=5*3/sqrt3*sqrt3/sqrt3`

`x=5sqrt3`

Then, plug-in `theta =30^o` and `x=5sqrt3` to EQ2.

`(d theta)/dt=(2000cos^2(30^o))/(5sqrt3)^2`

`(d theta)/dt=20`

Hence, when `theta =30^o` , the angle of elevation is changing at 20 radian per hour.

(b) `theta =60^o`

Plug-in the value of theta to EQ1 to get the value of x.

`x=5/tan (60^o)=5/sqrt3*sqrt3/sqrt3`

`x=(5sqrt3)/3`

Then, plug-in `theta =60^o` and `x=(5sqrt3)/3` to EQ2.

`(d theta)/dt=(2000cos^2(60^o))/((5sqrt3)/3)^2`

`(d theta)/dt=60`

Hence, when `theta =60^o` , the angle of elevation is changing at 60 radian per hour.

(c) `theta =80^o`

Plug-in the value of theta to EQ1 to get the value of x.

`x=5/tan (80^o)`

`x=0.8816`

Then, plug-in `theta =80^o` and `x=0.8816` to EQ2.

`(d theta)/dt=(2000cos^2(80^o))/(0.8816)^2`

`(d theta)/dt=77.59`

Hence, when `theta =80^o` , the angle of elevation is changing at 77.59 radian per hour.

Images:
This image has been Flagged as inappropriate Click to unflag
Image (1 of 1)

3 replies Hide Replies

nick-teal's profile pic

nick-teal | High School Teacher | (Level 3) Adjunct Educator

Posted on

You are exactly right!

There are 60 minutes in an hour.  So instead of 77.59 Radians per 1 hour, its 77.59 radians per 60 minutes.

So we divide 77.59 by 60 to get our answer in terms of radians per minutes.  1.293 Radians/Minute

jcamacho55's profile pic

jcamacho55 | Student, College Sophomore | (Level 2) eNoter

Posted on

Thank you so much for your help. One more question though. The answer is wanted in Radians per Minute. How would I convert the Answers from Radians Per Hour Into Radians per Minute? Is it simply dividing by 60? or is there more to it?

jcamacho55's profile pic

jcamacho55 | Student, College Sophomore | (Level 2) eNoter

Posted on

Like For Say, For the answer concerning 30 degrees. It's 20 radians per hour. would you basically divide 20/60 in order to achieve 1/3 radians per minute?

We’ve answered 318,916 questions. We can answer yours, too.

Ask a question