Question

If `A = 9/16(4r-sin(4r))` and `(dr)/dt=0.7` when `r=pi/4` then evaluate `(dA)/dt` when `r=pi/4`?

Answer 1

`[ (dA)/dt ]_(r=pi/4) = 3.15`

Explanation:

We have:

`A = (9/16)(4r-sin(4r))`

If we denote time in minutes by `t`, Then differentiating implicitly wrt `t` we have:

`\ \ \ (d)/dtA = (9/16)d/dt(4r-sin(4r))`

`:. (dA)/dt = (9/16)(dr)/dt d/(dr)(4r-sin(4r))`

`:. (dA)/dt = (9/16)(dr)/dt (4-4cos(4r))`

And we are given that `(dr)/dt=0.7`

`\ \ \ \ (dA)/dt = (9/4)(0.7) (1-cos(4r))`
`:. (dA)/dt = (1.575) (1-cos(4r))`

So, when `r=pi/4`, we have:

`[ (dA)/dt ]_(r=pi/4) = (1.575) (1-cos(pi))`
`" " = (1.575) (1-(-1))`
`" " = 3.15`