a pebble is dropped into a calm pond, causing ripples in the form of concentric circles. the radius r of the outer ripple is increasing at a constant rate of 1 root per second. When the radius is 4 feet, at what rate is the total area A of a disturbed water changing?
Thank you!
area=pi r^2
darea/dt=pi*2*r*dr/dt
is there something here you do not understand?
Fay/Grace/Alice/Mark --
You need to show some work on what you've posted. Otherwise, the math tutors will think you just want to copy the answers down.
To find the rate at which the total area A of the disturbed water is changing, we need to find the derivative of the area with respect to time t.
The area of a circle is given by the formula A = πr^2, where r is the radius.
Differentiating both sides of the equation with respect to time, we get:
dA/dt = d/dt (πr^2)
Using the chain rule, we get:
dA/dt = 2πr d(r)/dt
Given that the radius r is increasing at a constant rate of 1 foot per second, we have d(r)/dt = 1.
Substituting this value into the equation, we get:
dA/dt = 2π(4)(1)
Simplifying, we find:
dA/dt = 8π
Therefore, when the radius is 4 feet, the rate at which the total area of the disturbed water is changing is 8π square feet per second.
To find the rate at which the total area A of the disturbed water is changing, we need to use the chain rule of calculus. Let's first start by understanding the relationship between the radius and the area of the disturbed water.
The area A of a circle is given by the formula A = πr^2, where r is the radius. In this case, the radius is changing over time, so we need to consider how the area changes as the radius changes.
To do this, we can take the derivative of both sides of the equation with respect to time (t):
dA/dt = d(πr^2)/dt
Now, let's apply the chain rule. The chain rule states that if we have a function within a function, we need to multiply the derivative of the outer function with the derivative of the inner function.
In our case, the outer function is squaring the radius, and the inner function is the radius itself.
So, by applying the chain rule, we get:
dA/dt = 2πr * dr/dt
Now, we have the derivative of the area with respect to time in terms of the radius and the rate of change of the radius (dr/dt).
Given that the radius is increasing at a constant rate of 1 foot per second, dr/dt = 1 ft/s.
Now plug in the given radius value of 4 feet into the equation:
dA/dt = 2π * 4 * 1
Simplifying, we get:
dA/dt = 8π
So, the rate at which the total area A of the disturbed water is changing is 8π square feet per second.