Question

Question #1fa6b

Answer 1

Without knowing what the said angle is, we can only get a formula for answering this question.

Explanation:

We are interested in the Area of a right triangle, `A` and one of the acute angles, `theta`.

We are told that `(d theta)/dt = pi/16` rad / s and we are asked to find `(dA)/dt` when `theta` is _ _ _ . (not given).

We need an equation relating `A` and `theta`.

Sketch a right triangle with hypotenuse `40` cm and base angle `theta`

The area of a triangle is `1/2 "base" xx "height"`.

In my picture the base is the side adjacent to `theta`, so its length is `40cos theta`. The height is `40sintheta`.

The area is given by `A = 800 sintheta costheta` `"cm"^2`.

We now need to differentiate with respect to `t`. Rather than using the product rule for an implicit differentiation, let's rewrite the function first using `sin(2theta) = 2sintheta costheta`

`A =400sin(2theta)`

Now differentiate w.r.t. `t`.

`(dA)/dt = 800 cos(2theta) (d theta)/dt`

We were given `(d theta)/dt`, so we can write

`(dA)/dt = 50 pi cos(2 theta)`

Now, if we are given a `theta`, we can find the rate of change of the area.