Related rates finding the dA/dt.

[athamz]

Homework Statement

The measure of one of the acute angles of a right triangle is decreasing at the rate 1/36 pi rad/sec. If the length of the hypotenuse is constant at 40cm, find how fast the area is changing when the measure of the acute angle is 1/6 pi.

Homework Equations

The Attempt at a Solution

I used the formula for the area of a triangle which is 1/2 base times the height, using trigonometry function, I substitute the value of base with 40costeta and the height is 40sinteta. How will I differentiated A = 40costeta*40sinteta? Will I use the product rule to differentiate? where will I include or put the rate given?

 

[lanedance]

so you have
$$A(\theta) = 40 cos(\theta) sin(\theta)$$

differentiate both sides with respect to t, you will need to use both chain & product rules

 

[athamz]

oh.. meaning

$$dA/dt = (cos^2 [/theta] - sin^2 [/theta]) d[/theta]/dt$$

Where will I get the value for cos[/theta] and sin[/theta] ?

 

[lanedance]

well you're given theta

 

[athamz]

oh.. thank you so much..
Is it alright if I will get a negative answer?

 

[Char. Limit]

A note: This equation actually becomes very easy to differentiate if you recognize that $40 sin(\theta) cos(\theta) = 20 sin(2 \theta)$.

 

[lanedance]

i haven't checked if it is the correct answer, but assuming you've done the maths correctly a negative answer is fine, it just means the area is decreasing