Answer:
To find the rate at which the length of the third side of a triangle is increasing, we can use the Law of Cosines. The Law of Cosines states that in a triangle with sides of lengths a, b, and c, and the angle opposite side c denoted as θ, the following equation holds:
c^2 = a^2 + b^2 - 2ab*cos(θ)
In this case, we are given that one side has a length of 11 m (a) and another side has a length of 16 m (b). We want to find how fast the length of the third side (c) is changing with respect to time when the angle between the sides of fixed length is 60° and is increasing at a rate of 2°/min.
Let's denote the rate at which the angle is changing as dθ/dt. We are interested in finding dc/dt, the rate at which the length of the third side is changing.
Differentiating both sides of the Law of Cosines equation with respect to time (t), we get:
2c(dc/dt) = 2a(da/dt) + 2b(db/dt) - 2ab*sin(θ)*(dθ/dt)
Since da/dt and db/dt are both zero (as they represent sides with fixed lengths), and sin(θ) = sin(60°) = √3/2, we can simplify the equation to:
dc/dt = (ab*sin(θ)*(dθ/dt)) / c
Substituting the given values, we have:
a = 11 m
b = 16 m
θ = 60°
dθ/dt = 2°/min
Plugging these values into the equation, we get:
dc/dt = (11 * 16 * (√3/2) * (2°/min)) / c
Simplifying further:
dc/dt = (88√3 * (2°/min)) / c
dc/dt = (176√3°/min) / c
Now, we need to find the value of c when the angle θ is 60°. To do this, we can use the Law of Cosines equation again:
c^2 = a^2 + b^2 - 2ab*cos(θ)
c^2 = 11^2 + 16^2 - 2 * 11 * 16 * cos(60°)
c^2 = 121 + 256 - 176
c^2 = 201
c ≈ √201 ≈ 14.177 m
Substituting this value into the equation for dc/dt, we have:
dc/dt = (176√3°/min) / 14.177 m
dc/dt ≈ 8.249°/min
Therefore, the length of the third side is increasing at a rate of approximately 8.249°/min when the angle between the sides of fixed length is 60°.
Explanation: