two sides of triangle are 10m and 19m and the angle between them is increasing at a rate of 2 degrees per minute. How fast is the third side changing when the angle between them is 60 degrees
let the angle be Ø
given: dØ/dt = 2° per minute = 2(π/180) radians
= π/90 radians
(I switched to radians, since trig derivatives are valid for radians)
let the third side be x
x^2 = 10^2 + 19^2 - 2(10)(19)cosØ
x^2 = 461 - 380cosØ
2x dx/dt = 380sinØ dØ/dt
dx/dt = 190sinØ (π/90) / x
when x = 60° = π/3
x^2 = 461 - 380cosπ/3
= 271
x = √271
dx/dt = 190sin (π/3) (π/90) /√271
= .3489 m/min
Well, it seems like you're in quite the triangle pickle! Don't worry, I'm here to help you out with a smile.
To find out how fast the third side is changing, we can use the Law of Cosines. Let's call the third side of the triangle "c". The Law of Cosines states that c² = a² + b² - 2ab * cos(C), where "a" and "b" are the given sides and "C" is the angle between them.
In this case, the first side, "a", is 10m and the second side, "b", is 19m. We can start differentiating both sides with respect to time "t" to find the rate of change:
2c dc/dt = 2a da/dt + 2b db/dt - 2ab(-sin(C)) dC/dt
Applying the given information, da/dt = 0, db/dt = 0, and dC/dt = 2°/min, we can simplify the equation to:
2c dc/dt = 2ab(-sin(C)) dC/dt
Now, we know that C = 60° when we want to find the rate of change of the third side. Plugging in the values we have:
2c dc/dt = 2 * 10m * 19m * (-sin(60°)) * 2°/min
Simplifying further:
2c dc/dt = -2 * 10m * 19m * (√3/2) * 2°/min
Cancelling out:
c dc/dt = -380m² * (√3/2) * 2°/min
And finally:
dc/dt = (-380m² * (√3/2) * 2°/min) / c
Once you plug in the values for "c" in the triangle when C = 60°, you can determine the rate at which the third side is changing. Keep in mind, though, I'm just a clown bot, so you may want to double-check my calculations with your friendly neighborhood mathematician!
To find the rate at which the third side is changing, we can use the Law of Cosines.
The Law of Cosines states that in a triangle with sides a, b, and c, and angle C opposite side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab*cos(C)
Let's denote the first side as a = 10m, the second side as b = 19m, and the angle between them as C. We need to find how fast the third side, c, is changing, dc/dt, when C is 60 degrees and the rate at which C is changing, dC/dt, is 2 degrees per minute.
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 want to find dc/dt, we can rearrange the equation to solve for it:
dc/dt = (a * da/dt + b * db/dt - ab * sin(C) * dC/dt) / c
Given that a = 10m, b = 19m, C = 60 degrees, da/dt = 0 (because the length of side a is constant), db/dt = 0 (because the length of side b is constant), and dC/dt = 2 degrees per minute, we can substitute these values into the equation:
dc/dt = (10 * 0 + 19 * 0 - 10 * 19 * sin(60) * 2) / c
The sin(60) is equal to √3/2, so simplifying the equation further:
dc/dt = (-380√3) / c
To find the value of c, we can use the Law of Cosines:
c^2 = a^2 + b^2 - 2ab*cos(C)
c^2 = 10^2 + 19^2 - 2 * 10 * 19 * cos(60)
c^2 = 100 + 361 - 380
c^2 = 81
c = √81
c = 9m
Now, substituting c = 9m into the equation for dc/dt:
dc/dt = (-380√3) / 9
Simplifying further:
dc/dt = -20√3 m/min
Therefore, when the angle between the sides is 60 degrees, the third side of the triangle is changing at a rate of -20√3 meters per minute.
To find the rate at which the third side is changing, we can use the law of cosines, which relates the lengths of the sides of a triangle to the cosine of one of the angles. The law of cosines is given by:
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 opposite to side c.
In this case, we are given a = 10m and b = 19m. We want to find dc/dt, the rate at which the third side (c) is changing with respect to time (t), when the angle C is 60 degrees. We are also given dC/dt, the rate at which the angle C is changing with respect to time.
Let's differentiate both sides of the equation with respect to time:
2c * (dc/dt) = 2a * (da/dt) + 2b * (db/dt) - 2ab * sin(C) * (dC/dt)
Since we are given the rates da/dt, db/dt, and dC/dt, and we want to find dc/dt, we can substitute the given values into the equation and solve for dc/dt.
However, we are not given the values of da/dt and db/dt, which represent the rates at which the lengths of the other two sides are changing with respect to time. Without this information, we cannot determine the value of dc/dt.
Therefore, to solve this problem, we need to know the rates at which the lengths of the other two sides are changing.