Two sides of a triangle are 2 meters and 3 meters and the angle between them is increasing at 0.5 radians/second when the angle is pi/4.

Question

Two sides of a triangle are 2 meters and 3 meters and the angle between them is increasing at 0.5 radians/second when the angle is pi/4.

1. How fast is the distance between the tips increasing?

2. How fast is the area increasing?

I got part of #1 but I am getting multiple answers. Help me clear up the confusion. Thanks Mary!

Answer 1

use law of cosines to get the distance between tips.

c^2=2^2+3^2+2*2*3CosTheta

you are looking for dc/dt

2c dc/dt=-12sinTheta dTheta/dt

the fugure out dc/dt when theta=PI/4, and you will have to calculate c at that same angle.
I have no idea what you mean by "multiple " answers.

Answer 2

Mary, what part of yesterdays solution did you not like ?

http://www.jiskha.com/display.cgi?id=1456770767

Always check to see if your question has been answered before reposting the same thing.
That way we avoid unnecessary duplication of work.

Answer 3

To solve these problems, we can use the concepts of differentiation and related rates. Let's break down each problem:

1. How fast is the distance between the tips increasing?
To answer this, we need to find the rate of change of the distance between the tips (let's call it D) with respect to time. We can use the Law of Cosines to relate the sides of the triangle and the angle between them:

c^2 = a^2 + b^2 - 2ab*cos(theta)

In our case, a = 2 meters, b = 3 meters, and theta is initially pi/4 radians, which is increasing at a rate of 0.5 radians/second. So, we can differentiate the equation with respect to time:

2c * dc/dt = 2a * da/dt + 2b * db/dt - 2ab * dcos(theta)/dt

Since we are interested in finding the rate of change of D, we can rewrite the equation as:

2D * dD/dt = 2a * da/dt + 2b * db/dt - 2ab * dcos(theta)/dt

Given that a = 2 meters, b = 3 meters, and theta = pi/4 radians, we can substitute these values in and solve for dD/dt.

Now, you mentioned you got part of the answer, but you're getting multiple answers. To resolve this confusion, let's go through the calculation step by step:

a) Compute da/dt:
Since the length of side a remains constant, da/dt is zero.

b) Compute db/dt:
Similarly, the length of side b also remains constant, so db/dt is also zero.

c) Compute dcos(theta)/dt:
To find this, we need to differentiate the cosine function with respect to time and evaluate it at theta = pi/4. The derivative of cosine function is minus sine, so dcos(theta)/dt = -sin(theta). Substituting theta = pi/4, we get dcos(theta)/dt = -sin(pi/4) = -sqrt(2)/2.

d) Substitute the values in the equation:
Now, we can substitute the values we have computed into the equation:
2D * dD/dt = 2a * da/dt + 2b * db/dt - 2ab * dcos(theta)/dt
2D * dD/dt = 0 + 0 - 2 * 2 * 3 * (-sqrt(2)/2)
2D * dD/dt = 12 * sqrt(2)

e) Solve for dD/dt:
Finally, we can solve for dD/dt by dividing both sides of the equation by 2D:
dD/dt = 12 * sqrt(2) / (2D)

So, the speed at which the distance between the tips is increasing is given by this expression. Please note that we still need to know the specific value of D to compute the numerical answer.