Two sides of a triangle are 5 and 8 units respectively. If the included angle is changing at rate of one radian, at what rate is the third side changing when the included angle is 60 degrees?

One radian is not a rate. It is an angle. Look at the problem statement again.

To find the rate at which the third side of the triangle is changing, we can use the Law of Cosines.

The Law of Cosines states that for any triangle with sides a, b, and c, and angle C opposite side c:

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

Let's denote the third side of the triangle as "c" and the included angle as "C". We're given that side a has a length of 5 units, and side b has a length of 8 units.

When the included angle is 60 degrees, we need to convert it to radians. Since 180 degrees is equal to π radians, we can use the following conversion:

60 degrees * (π radians / 180 degrees) = π/3 radians.

Now, let's differentiate the equation with respect to time (t):

2c * (dc/dt) = 2a * (da/dt) + 2b * (db/dt) - 2ab * (-sin(C) * (dC/dt))

Since we're given that the included angle is changing at a rate of one radian:

(dC/dt) = 1

Substituting the given values:

2c * (dc/dt) = 2(5) * (da/dt) + 2(8) * (db/dt) - 2(5)(8) * (-sin(π/3))

2c * (dc/dt) = 10(da/dt) + 16(db/dt) + 80 * (√3 / 2)

2c * (dc/dt) = 10(da/dt) + 16(db/dt) + 40√3

Now, we need to find the values of (da/dt) and (db/dt). We're not given any explicit information about their rates of change, so we cannot determine them without further information.

Hence, we cannot determine the rate at which the third side of the triangle is changing without additional information.

To find the rate at which the third side is changing, we can use the law of cosines, which relates the sides and the included angle of a triangle. The law of cosines states:

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

where c is the third side (which we are trying to find the rate of change for), a and b are the given sides, and C is the included angle.

Let's denote the third side as c(t) and the included angle as C(t), where t is time.

We are given that the included angle, C, is changing at a rate of one radian per unit time. Therefore, the rate of change of the included angle is dC/dt = 1 rad/unit time.

We want to find dc/dt, the rate at which the third side is changing.

Now, to find the rate of change of the third side, we can differentiate both sides of the law of cosines equation with respect to time t:

2c(t) * dc/dt = 2a * da/dt + 2b * db/dt - 2ab * sin(C) * dC/dt.

Since we want to find dc/dt, we rearrange the equation and solve for it:

dc/dt = (a * da/dt + b * db/dt - ab * sin(C) * dC/dt) / c(t).

To evaluate this equation at the given time when the included angle is 60 degrees, we need to convert 60 degrees to radians. Recall that π radians = 180 degrees, so 60 degrees is (60/180) * π radians.

Now, we substitute the given information from the problem into the equation:

a = 5 units,
b = 8 units,
C = (60/180) * π radians,
da/dt = db/dt = 0 (since the lengths of the sides are constant),
dC/dt = 1 rad/unit time.

Substituting these values into the equation, we have:

dc/dt = (5 * 0 + 8 * 0 - 5 * 8 * sin((60/180) * π) * 1) / c(t).

Simplifying the equation:

dc/dt = -40 * sin((60/180) * π) / c(t).

Therefore, the rate at which the third side is changing when the included angle is 60 degrees is -40 * sin((60/180) * π) / c(t).

drwls is right, you must have left out the units of time.

(e.g. ... is changing at a rate of one radian per minute )

In that case, use the cosine law equation as your starting point

let the third side be x and the angle be Ø

x^2 = 5^2 + 8^2 - 2(5)(8)cosØ

2x dx/dt = 0 + 0 + 80 sinØ dØ/dt

from the original equation, find x , then sub in the given values

pretty easy after that.