A stone dropped into water produces a circular ripple which increases in radius at a rate of 1.50 m/s. How fast is the area increasing within the ripple when its diameter is 6.00m?
a = pir^2
da/dt = 2pi*r dr/dt
da/dt = 2pi*3 (1.5) = 9pi
To find the rate at which the area of the circular ripple is increasing, we need to differentiate the area equation and substitute the given values.
The area of a circle is given by the formula:
A = πr², where r is the radius of the circle.
We are given that the radius of the circular ripple is increasing at a rate of 1.50 m/s. Therefore, the rate of change of the radius with respect to time is: dr/dt = 1.50 m/s.
We want to find dA/dt, the rate of change of the area with respect to time. To do this, we need to differentiate the area formula with respect to time:
dA/dt = d(A)/dt = d(πr²)/dt
Using the chain rule, we can differentiate this equation by treating r as a function of time:
dA/dt = d(πr²)/dt = 2πr(dr/dt)
Substituting the given values, we have:
r = 6.00 m (diameter is 6.00 m)
dr/dt = 1.50 m/s
dA/dt = 2π * 6.00 m * (1.50 m/s)
= 18π m²/s
The rate at which the area of the circular ripple is increasing when its diameter is 6.00 m is 18π m²/s.
To find the rate at which the area is increasing within the ripple, we need to use the related rates formula.
Let's denote the radius of the ripple as r and the time as t. We are given that the rate at which the radius increases, dr/dt, is 1.50 m/s. We are also given that the diameter of the ripple is 6.00 m, which means the radius is half of that or 3.00 m.
Now, let's relate the variables using the formula for the area of a circle: A = π * r^2.
Taking the derivative with respect to time (t) of both sides of the equation, we get:
dA/dt = d/dt(π * r^2)
The left side represents the rate at which the area is increasing, which is what we want to find.
Now, let's differentiate the right side of the equation using the chain rule:
dA/dt = d/dt(π * r^2) = 2πr * dr/dt
Since we know the values of r (3.00 m) and dr/dt (1.50 m/s), we can substitute them into the equation:
dA/dt = 2π * 3.00 m * 1.50 m/s
Simplifying this expression, we find:
dA/dt = 9π m^2/s
Therefore, the area is increasing at a rate of 9π m^2/s within the ripple when its diameter is 6.00 m.