A rock is thrown in a pond, and creates circular ripples whose radius increases at a rate of 0.2 meter per second. What will be the value of dA/dt where A is the area (in square meter) of the circle after 5 seconds?
A = πr^2
dA/dt = 2πr dr/dt
= ....
You are given dr/dt and
you know that after 5 seconds the radius will be 5s * .2m/s = 1 m
so just sub it in.
@mathhelper
I got 0.4pi m^2/sec
Well, let's see. We know that the radius of the circle increases at a rate of 0.2 meters per second. So, after 5 seconds, the radius would have increased by 5 times 0.2, which is 1 meter. Now, the area of a circle is given by the formula A = πr^2. Since the radius is now 1 meter, the area would be π times 1 squared, which is π square meters. But, hold on a second. I just realized that you asked for the value of dA/dt, which represents the rate of change of the area with respect to time. In this case, since the radius is increasing at a constant rate, the area is also increasing at a constant rate. Therefore, the value of dA/dt would be 0. So, after all that, the answer is... zero!
Given that the radius of the circle increases at a rate of 0.2 meters per second, we can find the rate of change of the area of the circle using the formula for the area of a circle:
A = πr²
Taking the derivative of both sides of the equation with respect to time (t), we get:
dA/dt = d(πr²)/dt
Using the chain rule, the derivative of r² with respect to t is:
d(πr²)/dt = 2πr(dr/dt)
We know that dr/dt, the rate of change of the radius, is 0.2 meters per second. So substituting this into the equation, we have:
dA/dt = 2πr(0.2)
To find the value of dA/dt after 5 seconds, we need to know the radius of the circle at that time. Since we haven't been provided with the initial radius of the circle, we cannot find the exact value of dA/dt.
To find the value of dA/dt, we need to first determine the equation that relates the area of the circle to its radius. The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius.
We are given that the radius is increasing at a rate of 0.2 meters per second. So, the rate of change of the radius with respect to time, dr/dt, is 0.2 m/s.
Using the chain rule of differentiation, we can express the rate of change of the area with respect to time, dA/dt, in terms of dr/dt:
dA/dt = dA/dr * dr/dt
To find dA/dr, we differentiate the equation for the area of a circle with respect to the radius:
dA/dr = 2πr
Now we can substitute the given value of dr/dt and find the value of dA/dt:
dA/dt = (2πr) * (0.2 m/s)
To evaluate dA/dt after 5 seconds, we need to know the radius of the circle at that time. Since the radius increases at a constant rate of 0.2 meters per second, after 5 seconds, the radius will have increased by 5 * 0.2 = 1 meter.
Plugging in r = 1 meter into the equation for dA/dt:
dA/dt = (2π * 1) * (0.2 m/s) = 0.4π m²/s
Therefore, the value of dA/dt, after 5 seconds, is 0.4π square meters per second.