A stone is dropped into the sea, causing circular waves whose radii increases at a constant rate of 1.5 per minute. At what rate is the circumference of the wave is changing?
Thank you!
Circumderence=2PI*r
d(Circum)/dt=2PI*dr/dt
To find the rate at which the circumference of the wave is changing, we can use the formula for the circumference of a circle.
Circumference (C) = 2 * π * r
where C is the circumference and r is the radius.
Since the radius is increasing at a constant rate of 1.5 units per minute, we need to find the rate of change of the circumference with respect to time.
To do this, we can differentiate the equation with respect to time (t):
dC/dt = 2 * π * dr/dt
where dC/dt is the rate of change of the circumference with respect to time, and dr/dt is the rate of change of the radius with respect to time.
In this case, dr/dt is given as 1.5 units per minute.
Substituting these values into the equation, we get:
dC/dt = 2 * π * 1.5
Simplifying the equation, we have:
dC/dt = 3π
Therefore, the rate at which the circumference of the wave is changing is equal to 3π units per minute.
To find the rate at which the circumference of the wave is changing, we can use the formula for the circumference of a circle:
C = 2πr
Where C is the circumference and r is the radius of the circle.
We are given that the radius is increasing at a constant rate of 1.5 units per minute. We need to find the rate at which the circumference is changing, or dC/dt.
To do this, we can differentiate the equation for the circumference with respect to time:
dC/dt = d/dt(2πr)
Since r is changing with time, we need to use the chain rule:
dC/dt = 2π(d/dt(r))
The derivative of r with respect to time is given as 1.5 units per minute. So we plug in this value:
dC/dt = 2π(1.5)
Simplifying this expression, we get:
dC/dt = 3π
So, the rate at which the circumference of the wave is changing is 3π units per minute.