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.
Please help I have been stuck on this problem for days!
Just use the cosine law
let x be the distance between them when the angle is Ø
x^2 = 8^2 + 13^2 - 2(8)(13)cosØ
x^2 = 233 - 208cosØ
2x dx/dt = 208sinØ dØ/dt
dx/dt = (104sinØ dØ/dt)/x
given: dØ/dt = .08 rads/min
find : dx/dt when Ø = π/3
we will need x for the case when Ø = π/3.
x^2 = 233 - 208cosπ/3 = 129
x = √129
dx/dt = 104sinπ/3 (.08)/√129
= ....
if get appr. .634 m/min
check my arithmetic
To solve this problem, we can use the law of cosines to relate the lengths of the sides of the triangle to the angle between them. The law of cosines states that for any triangle with sides of lengths a, b, and c, and with the angle between sides a and b denoted as θ, the following equation holds true:
c^2 = a^2 + b^2 - 2ab*cos(θ)
In this case, we are given the lengths of sides a and b (8 m and 13 m, respectively) and the rate at which the angle θ is increasing (0.08 radians/minute). We want to find how fast the length of side c is changing when θ = π/3 radians.
To find the rate at which side c is changing, we can take the derivative of both sides of the equation above with respect to time:
d/dt(c^2) = d/dt(a^2 + b^2 - 2ab*cos(θ))
2c * dc/dt = 2a * da/dt + 2b * db/dt + 2ab * d(θ)/dt * sin(θ)
Since we are given that da/dt = 0, db/dt = 0, and d(θ)/dt = 0.08 radians/minute, and we want to find dc/dt, we can simplify the equation to:
2c * dc/dt = 2ab * d(θ)/dt * sin(θ)
Now, we can substitute the given values into the equation:
2c * dc/dt = 2(8 m)(13 m) * (0.08 radians/minute) * sin(π/3 radians)
Simplifying further:
2c * dc/dt = 208 * 0.08 * (sqrt(3)/2)
2c * dc/dt = 16.64 * (sqrt(3)/2)
2c * dc/dt = 28.8
Dividing both sides of the equation by 2c:
dc/dt = 14.4 / c
Now, we need to find the value of c when θ = π/3 radians. To do this, we can use the law of cosines again:
c^2 = a^2 + b^2 - 2ab*cos(π/3)
c^2 = 8^2 + 13^2 - 2(8)(13)*cos(π/3)
c^2 = 64 + 169 - 208*cos(π/3)
c^2 = 233 - 208*(1/2)
c^2 = 233 - 104
c^2 = 129
Taking the square root of both sides:
c ≈ sqrt(129) ≈ 11.36
Now we can substitute the value of c into the equation we derived earlier:
dc/dt ≈ 14.4 / 11.36
dc/dt ≈ 1.268 meters/minute
Therefore, when the angle between the sides of fixed length is π/3 radians, the length of the third side is increasing at a rate of approximately 1.268 meters/minute.