Two sides of a triangle have lengths 8 m and 13 m. The angle between them is increasing at a rate of 0.08 radians /min. How fast is the length of the third side increasing when the angle between the sides of fixed length is π/3 radians.
To find the rate at which the length of the third side is increasing, we can use the law of cosines. The law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.
In this case, let's denote the length of the third side as c, and the angle between the sides of length 8m and 13m as θ.
The law of cosines states that:
c^2 = a^2 + b^2 - 2ab * cos(θ)
where a and b are the lengths of the other two sides of the triangle.
In our case, a = 8m, b = 13m, and the angle θ is increasing at a rate of 0.08 radians/minute.
To find how fast the length of the third side c is increasing, we need to take the derivative of both sides of the equation with respect to time (t):
2c(dc/dt) = 2a(da/dt) + 2b(db/dt) - 2ab(d(cos(θ))/dt)
The term (dc/dt) represents how fast the length of the third side is increasing.
However, we are given that the angle θ is π/3 radians when finding the rate at which the length of the third side is increasing.
Therefore, we can substitute a = 8m, b = 13m, θ = π/3 into the equation.
To find the rate of change of the length of the third side, we can use the Law of Cosines, which relates the length of the third side to the lengths of the other two sides and the angle between them.
The Law of Cosines states:
c^2 = a^2 + b^2 - 2ab * cos(C)
Where c is the length of the third side, a and b are the lengths of the other two sides, and C is the angle between sides a and b.
In this case, we have a = 8 m, b = 13 m, and dC/dt = 0.08 radians/minute, where C is the angle between the sides of fixed length. We need to find dc/dt, the rate at which the length of the third side is changing.
Differentiating both sides of the Law of Cosines with respect to time t, we get:
2c(dc/dt) = 2a(da/dt) + 2b(db/dt) - 2ab * (-sin(C) * dC/dt)
Since da/dt and db/dt are both zero (the lengths of the sides are fixed), the equation simplifies to:
2c(dc/dt) = 2ab * sin(C) * (dC/dt)
Now we can substitute the given values:
c = unknown (length of the third side)
a = 8 m
b = 13 m
C = π/3 radians
dC/dt = 0.08 radians/minute
Plugging in these values, we have:
2c(dc/dt) = 2 * 8 * 13 * sin(π/3) * (0.08)
Simplifying further:
2c(dc/dt) = 208 * sin(π/3) * (0.08)
c(dc/dt) = 104 * √3 * 0.08
c(dc/dt) = 8.384 * √3
Therefore, the rate at which the length of the third side is increasing when the angle between the sides of fixed length is π/3 radians is approximately 8.384 * √3 meters per minute.
pi/3 = 60 degree just so I do not have to use radians on my calculator :)
c^2 = a^2 + b^2 - 2 a b cos C
get initial stuff first
cos 60 = 1/2
sin 60 = .866
c^2 = 64 + 169 - 208 cos 60
c^2 = 233 - 104 = 129
c = 11.4
now do the problem
2 c dc/dt = -208 (-sin 60) dC/dt
so
dc/dt = 208(.866)(.08)/(2*11.4)