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A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 70 cm/s. Find the rate (in cm2/s) at which the area within the circle is increasing after each of the following.

(a) after 1 s

User Shigeo
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1 Answer

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Let's start by using the formula for the area of a circle:

A = πr^2

where A is the area and r is the radius.

We are given that a circular ripple is created in a lake and it travels outward at a speed of 70 cm/s. This means that the radius of the circle is increasing at a rate of 70 cm/s.

(a) After 1 second, the radius of the circle is:

r = 70 cm/s × 1 s = 70 cm

Now we can use the formula for the area of a circle to find the rate at which the area within the circle is increasing:

A = πr^2

A = π(70 cm)^2

A ≈ 15,400 cm^2

To find the rate at which the area within the circle is increasing, we need to take the derivative of the area with respect to time:

dA/dt = 2πr(dr/dt)

where dA/dt is the rate at which the area is increasing, and dr/dt is the rate at which the radius is increasing (which we know is 70 cm/s).

Plugging in the values we have:

dA/dt = 2π(70 cm)(70 cm/s)

dA/dt ≈ 9,800 cm^2/s

So after 1 second, the area within the circle is increasing at a rate of approximately 9,800 cm^2/s.
User Jesse S
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