<p>Hi guys, our teacher gave us this huge worksheet of related rates problems but I'm having trouble answering one of them...Can anyone help? By the way, the answer is posted after the question, but I just need to know HOW to solve it. Thanks in advance!</p>
<p>In triangle ABC, AB = 15 in and AC = 20 in. The angle between AB and BC is increasing at pi/90 radians per second. How fast is the length of the third side changing when the angle between the sides is pi/3 radians.</p>
<p>Answer: (pi)/(sqrt(39)) in/sec</p>
<p>Okay, so first we draw a model. You have a triangle ABC with side AC=20 in. and AB=15 in. and angle theta between sides AB and AC. To start with, we need an equation for the side opposite angle A, because we want to find how fast that side is changing.
Using law of cosines, a^2 = b^2 + c^2 - 2bc<em>cos(theta).
Substitute 20 and 15 for b and c: a^2 = 625 - 600cos(theta).
Differentiate both sides of the equation: 2a</em>da/dt = 600sin(theta).
Solve for da/dt: da/dt = (300sin(theta)dtheta/dt)/a
we know what theta equals: pi/3
Now find a: a = sqrt(625 - 600cos(pi/3)) = sqrt(325) = 5<em>sqrt(13)
now plug in the numbers in the equation we got earlier: da/dt = 300sin(theta)/a:
da/dt = (300sin(pi/3)</em>pi/90)/(5sqrt(13)) = pi/sqrt(39)</p>
<p>hope this helps</p>
<p>OMG, Thankyou sooo much, I had no idea about using law of cosines. :-)</p>