Two sides of a triangle are 5 and 8 and the angle between them is increasing at .05 rad/sec. how fast is the distance between the tips of the sides increasing when the angle is pi/4
let that distance be x
angle be Ø
by the cosine law:
x^2 = 25+64-2(5)(8)cosØ
x^2 = 89 - 80cosØ
2x dx/dt = 80sinØ dØ/dt
dx/dt = 40sinØ dØ/dt / x ***
when Ø = π/4 or 45°
x^2 = 89-80sin π/4
= 32.431...
x = 5.6948... (I stored that in memory)
back in ***
dx/dt = 40(sin (π/4)) (-.05)/5.6948...
= -.24833.. units per second
(the - signs shows that the distance is decreasing)
To determine how fast the distance between the tips of the sides is increasing, we can use the Law of Cosines to find the length of the third side of the triangle. Then, we can apply the chain rule to calculate the rate of change of the distance between the tips of the sides with respect to time.
Let's start by using the Law of Cosines to find the length of the third side of the triangle. The Law of Cosines states that:
c^2 = a^2 + b^2 - 2ab*cos(C)
Given that the two sides of the triangle are 5 and 8, and the angle between them is increasing at 0.05 rad/sec, we can substitute these values into the equation. Let's consider the angle between the sides as θ.
c^2 = 5^2 + 8^2 - 2*5*8*cos(θ)
Now, we need to find how fast the distance between the tips of the sides is increasing with respect to time. Let's denote the distance as D(t), and time as t.
We can determine the relationship between the distance D(t) and the sides of the triangle using the Cosine rule, which states:
D(t)^2 = a^2 + b^2 - 2ab*cos(θ)
As the angle θ is changing with time t, we need to differentiate the equation with respect to time t using the chain rule.
2*D(t)*d(D(t))/dt = 2*a*da/dt + 2*b*db/dt - 2*ab*sin(θ)*dθ/dt
Since we are interested in finding the rate of change of the distance between the tips of the sides, d(D(t))/dt, we rearrange the equation to solve for it:
d(D(t))/dt = (a*da/dt + b*db/dt - ab*sin(θ)*dθ/dt) / D(t)
Now, let's substitute the given values and the value of θ = π/4 radians into the equation.
a = 5, da/dt = 0 (as side a is constant)
b = 8, db/dt = 0 (as side b is constant)
θ = π/4
dθ/dt = 0.05 rad/sec
Using these values, we can calculate:
d(D(t))/dt = (5*0 + 8*0 - 5*8*sin(π/4)*0.05) / D(t)
= - 5 * 8 * sin(π/4) * 0.05 / D(t)
Now, we need to find the length of the third side, D(t), to calculate the final result. Using the Law of Cosines, we substitute θ = π/4 radians:
D(t)^2 = 5^2 + 8^2 - 2*5*8*cos(π/4)
= 25 + 64 - 80/sqrt(2)
= 89.141
Taking the square root of both sides, we get:
D(t) = sqrt(89.141)
= 9.43 (approx.)
Now, we can substitute this value into the equation for d(D(t))/dt:
d(D(t))/dt = - 5 * 8 * sin(π/4) * 0.05 / 9.43
= - 5 * 8 * sin(π/4) * 0.05 / 9.43
= - 2 * sqrt(2) * 0.05 / 9.43
= - 0.014 (approx.)
Therefore, the distance between the tips of the sides is decreasing at a rate of approximately -0.014 units per unit of time, when the angle is π/4 radians.