Calculus question

Question

The height h of a right triangle is increasing at a rate of 1 unit/s. Find the rate at which theta (in rad/s) is changing when h =8.

Help!?

Answer 1

 Angle Theta = T is a function of time t

 Height H is also a function of time t.

 The other leg of the right triangle is a fixed number constant C.

 

    tan(T) = H/C = (1/C)*H

 

   We seek dT/dt

 

    Implicit Differentiation:

 

    sec(T)^2 * dT/dt = 1/C* (dH/dt)  <--- 1st derivative.

 

    When the height H=8, then tan(T) = 8/C

   T = arctan(8/C)

 

   sec( arctan(8/C))^2 * dT/dt = 1/C*dH/dt

 

When the height H=8, the cosine of T is cos(T) = C/(sqrt(C^2 + 64))

 

so the sec(T) = sqrt(C^2 + 64)/C

 

sec(T)^2 = (C^2+64)/C^2

 

solving the first derivative for dT/dt and plugging in:

 

  dT/dt = 1/C * C^2/(C^2 + 64)

 

Note here that we are dividing by a fraction; so it gets flipped;

remember when dividing fractions KFC: keep, flip, change.

Also dH/dt = 1, so I just ignored it as it multiplies out

 

The final result is C/(C^2 + 64) as the C^2 cancels the C

 

So when we have the base measure of the right triangle that remains a fixed constant,

plug it in to find the speed of the changing angle