Question

The equation

+2 [cos(30)
9 cos(6w) = 4+2
defines an implicit relationship between wand 8. Use implicit
dw
differentiation to find a formula for in terms of w and . The formula
do
will have the form
dw
de
where K is a constant. What is the value of K?
=
= Kcos(30) sin(30)/sin(6w)

Answer 1

`\sf K = 0.222 \textsf{(in 3 decimal places)}`

Explanation:

To find
`\sf (dw)/(d\theta)` using implicit differentiation, let's start by differentiating both sides of the given equation with respect to
`\sf \theta`:

`\sf 9\cos(6w) = 4 + 2[\cos(3\theta)]^2`

Differentiate both sides with respect to
`\sf \theta`:

`\sf (d)/(d\theta)\left(9\cos(6w)\right) = (d)/(d\theta)\left(4 + 2[\cos(3\theta)]^2\right)`

Now, apply the chain rule on the left side and the power rule on the right side:

`\sf 9(d)/(d\theta)\cos(6w) = 0 + 2 * 2 \cos(3\theta)(d)/(d\theta)\cos(3\theta)`

For the left side, use the chain rule:
`\sf (d)/(d\theta)\cos(6w) = -6\sin(6w)(dw)/(d\theta)`:

`\sf -54\sin(6w)(dw)/(d\theta) = 4\cos(3\theta)(-3\sin(3\theta))`

`\sf -54\sin(6w)(dw)/(d\theta) = -12\cos(3\theta)\sin(3\theta)`

Now, solve for
`\sf (dw)/(d\theta)`:

`\sf (dw)/(d\theta) = (12\cos(3\theta)\sin(3\theta))/(54\sin(6w))`

Simplify the expression:

`\sf (dw)/(d\theta) = (2\cos(3\theta)\sin(3\theta))/(9\sin(6w))`

Now, compare this with the given form:

`\sf (dw)/(d\theta) =( K\cos(30)\sin(30))/(sin(6w))`

Comparing coefficients, we have:

`\sf K = (2)/(9)`

So, the value of
`\sf K` is
`\sf (2)/(9)`, and as a decimal rounded to three decimal places, it is approximately
`\sf 0.222`.