# You throw a rock into a pond and watch the circular ripple travel out in all directions along the surface. If the ripple travels at 1.4 m/s, what is the approximate rate that the circumference is increasing when the diameter of the circular ripple is 6m?

2.8pi m/s

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To find the rate that the circumference is increasing when the diameter of the circular ripple is 6m, you can use the formula for the circumference of a circle: C = π * d, where C is the circumference and d is the diameter.

Given that the diameter is 6m, substitute it into the formula to find the circumference. Then, differentiate the circumference formula with respect to time to find the rate of change of the circumference. Since the radius of the circle is half of the diameter, the radius is 3m.

Differentiate the circumference formula with respect to time, you get:

dC/dt = π * (2 * dr/dt)

Given that the ripple travels at 1.4 m/s, and the rate of change of radius (dr/dt) is the same as the speed of the ripple, which is 1.4 m/s.

Substitute the values into the formula:

dC/dt = π * (2 * 1.4)

Calculate:

dC/dt ≈ π * 2 * 1.4

dC/dt ≈ 8.8π

So, the approximate rate that the circumference is increasing when the diameter of the circular ripple is 6m is approximately 8.8π meters per second.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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