8. Two motorcycles, travelling at the same speed, approach an intersection, one from the north and the other from the west.
When one motorcycle is 110 m north and
the other is 80 m west of the intersection,
the distance between them is decreasing
at 30 m/s. Determine the speed of the motorcycles
n^2 + w^2 = d^2 ... d = √(n^2 + w^2)
differentiating ... 2 n dn/dt + 2 w dw/dt = 2 d dd/dt
dn/dt = dw/dt = s
2 * 110 * s + 2 * 80 * s = 2 * √(110^2 + 80^2) * -30
s is negative because the motorcycles are approaching the intersection
the distance z is
z^2 = x^2 + y^2
z dz/dt = x dx/dt + y dy/dt
since they have the same speed s, that means
z dz/dt = s(x+y)
at (80,110), that gives
10√185 (-30) = 190s
now finish it off
what does it means that s is negative?
To determine the speed of the motorcycles, we need to look at their relative motion and the rate at which the distance between them is decreasing.
Let's assume that the motorcycle coming from the north has a speed of v₁ m/s, and the motorcycle coming from the west has a speed of v₂ m/s. Since they are both traveling at the same speed, v₁ = v₂ = v.
Given that the distance between the motorcycles is decreasing at a rate of 30 m/s, we can set up an equation using the concept of relative motion.
The distance between the motorcycles at any given time can be calculated using the Pythagorean theorem:
d = √((110 - v₁t)² + (80 - v₂t)²)
Taking the derivative of this equation with respect to time (t), we can find the rate of change of the distance between them:
d' = -2(110 - v₁t)v₁ - 2(80 - v₂t)v₂
Since the rate of change of the distance is given as -30 m/s, we can set d' equal to -30 and solve for t:
-2(110 - vt)v - 2(80 - vt)v = -30
Simplifying this equation, we get:
-220v + 2v²t - 160v + 2v²t = -30
Combining the v²t terms, we have:
4v²t - 380v + 30 = 0
Now, we can solve this quadratic equation for v, the speed of the motorcycles.
It's important to note that the equation and its solution depend on the specific values given for the distances and rates. Without those values, we can't determine the exact speed of the motorcycles.