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A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 40 cm/s. Find the rate at which the area within the circle is increasing after each of the following.
(a) after 2 s
cm2/s
(b) after 5 s
cm2/s
(c) after 6 s
cm2/s
To find the rate at which the area within the circle is increasing, we need to use the formula for the area of a circle:
A = πr^2
where A is the area and r is the radius of the circle.
In this case, the radius of the circle is increasing at a constant rate as the ripple travels outward. This means that the rate of change of the radius, dr/dt, is constant.
To find the rate at which the area is increasing, we need to differentiate the formula for the area of a circle with respect to time (t), using the chain rule. The chain rule states that if y = f(u) and u = g(x), then dy/dx = (dy/du)(du/dx).
Differentiating A = πr^2 with respect to t, we get:
dA/dt = (dA/dr)(dr/dt)
Since A = πr^2, the derivative dA/dr is:
dA/dr = 2πr
The derivative dr/dt is given as 40 cm/s.
Now, we can calculate the rate at which the area is increasing after each of the given time intervals:
(a) After 2 s:
To calculate this, we need to know the initial radius of the ripple. Let's assume it is r0. Therefore, after 2 seconds, the radius of the circle will be r = r0 + (40 cm/s)(2 s) = r0 + 80 cm.
The rate at which the area is increasing after 2 seconds is:
dA/dt = (dA/dr)(dr/dt)
= (2πr) (40 cm/s)
= 2π(r0 + 80 cm)(40 cm/s)
(b) After 5 s:
Following similar steps, after 5 seconds, the radius becomes r = r0 + (40 cm/s)(5 s) = r0 + 200 cm.
The rate at which the area is increasing after 5 seconds is:
dA/dt = (dA/dr)(dr/dt)
= (2πr) (40 cm/s)
= 2π(r0 + 200 cm)(40 cm/s)
(c) After 6 s:
Using the same approach, after 6 seconds, the radius becomes r = r0 + (40 cm/s)(6 s) = r0 + 240 cm.
The rate at which the area is increasing after 6 seconds is:
dA/dt = (dA/dr)(dr/dt)
= (2πr) (40 cm/s)
= 2π(r0 + 240 cm)(40 cm/s)
Now, you can substitute the value of r0 and calculate the respective rates of increase after 2 s, 5 s, and 6 s.