MHB How Fast Does the Area of a Triangle Change with Increasing Angle?

[tmt1]

Two sides of a triangle are 4 m and 5 m in length and the angle between them is increasing at a rate of 0.06 rad/s. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is $$\pi/3$$

When the angle is $$pi/3$$, the third side is equal to : $$\sqrt{21}$$ based on the law of cosines.I'm not sure which formula for Area I should use to figure this problem out.

 

[Sudharaka]

Two sides of a triangle are 4 m and 5 m in length and the angle between them is increasing at a rate of 0.06 rad/s. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is $$\pi/3$$

When the angle is $$pi/3$$, the third side is equal to : $$\sqrt{21}$$ based on the law of cosines.I'm not sure which formula for Area I should use to figure this problem out.

Hi tmt, :)

You could use derivatives to solve this question. Suppose the angle between the two given sides is $\theta$. Then the area of the triangle is given by,

\[A=\frac{4\times 5\times \sin\theta}{2}\]

You have to find $\frac{dA}{dt}$ when $\theta=\frac{\pi}{3}$. Try to do it from here. :)

 

[tmt1]

Hi tmt, :)

You could use derivatives to solve this question. Suppose the angle between the two given sides is $\theta$. Then the area of the triangle is given by,

\[A=\frac{4\times 5\times \sin\theta}{2}\]

You have to find $\frac{dA}{d\theta}$ when $\theta=\frac{\pi}{3}$. Try to do it from here. :)

\[A=\frac{4\times 5\times \sin\theta}{2}\]So I differentiate this with respect to time in seconds.

$\d{A}{t} = \frac{1}{2} \cos\theta \d{\theta}{t}$

I plug this in:

$\cos \pi / 2 = 1/2$

and I know that

\d{\theta}{t} = 0.06 radians/second

so

$\d{A}{t} = 0.06/ 4 m^2 / second$

or

0.015 squared/ second, but the answer in the text is 0.3 metres squared a second?

 

[SamJohannes]

$\d{A}{t} = \frac{1}{2} \cos\theta \d{\theta}{t}$

Double check this one. You've left out the side lengths from your derivative

 

[tmt1]

Double check this one. You've left out the side lengths from your derivative

The side length are constant so when they are differentiated they go to 0, don't they?

 

[MarkFL]

The side length are constant so when they are differentiated they go to 0, don't they?

Not when they are factors...recall:

$$\d{}{x}\left(k\cdot f(x)\right)=k\cdot\d{}{x}\left(f(x)\right)$$