Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

A kite is released from a spot 50 feet from where you are on a calm day. It rises at a rate of 3 ft/s. How fast is the angle of elevation changing when it is $\frac{\pi}{6}$ radians above the line of sight from where you are? What about when the angle is $\frac{\pi}{4}$?

By creating a triangle, I can tell that the kite will be 25 feet in the air but I'm unsure how to incorporate the rate. Can someone walk me through the problem?

Just a heads up, this isn't a homework problem; rather it is a problem I got wrong on an exam.

share|improve this question

1 Answer 1

up vote 2 down vote accepted

Let $D$ be the horizontal distance to the kite, and $H$ the height of the kite above the ground. Then $\tan \theta = H/D$, where $\theta$ is the angle of elevation from where you're standing. Now take $d / dt$ of this equation, noting that $D$ is constant: $$ \frac{d}{dt}\tan \theta = \frac{d}{dt} \left(\frac{H}{D}\right) \Rightarrow \sec^2 \theta \frac{d\theta}{dt} = \frac{1}{D} \frac{dH}{dt} \Rightarrow \frac{d\theta}{dt} = \frac{\cos^2 \theta}{D} \frac{dH}{dt} $$ We know $D = 50$ ft and $dH / dt = 3$ ft/s, so you just have to substitute $\theta$ to get $d\theta / dt$.

share|improve this answer
    
So that would make $\frac{\pi}{6}$ at .045 degrees per second? I wish I had known it was this simple to set up the equation :/ I missed the obvious –  StrugglingWithMath Dec 5 '12 at 19:32
    
0.045 radians per second. Multiply by 180/$\pi$ to get degrees per second. –  Eric Angle Dec 5 '12 at 22:09

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.