# The radius r of a sphere is increasing at a constant rate of 0.04 centimeters per second.At the time when the radius of the sphere is 10 centimeters, what is the rate of increase of its volume?

The rate of increase would be

The constant rate of the radius increase is 0.04 cm per second, which can be written as:

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When the radius of the sphere is 10 centimeters, the rate of increase of its volume can be found using the formula for the volume of a sphere:

[V = \frac{4}{3} \pi r^3]

Taking the derivative of both sides with respect to time:

[\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}]

Substituting the given values:

[r = 10 , cm] [\frac{dr}{dt} = 0.04 , cm/s]

[r = 10] [\frac{dr}{dt} = 0.04]

[\frac{dV}{dt} = 4\pi (10)^2 (0.04)]

[= 16\pi \times 0.04 \times 100]

[= 16\pi , cm^3/s]

So, the rate of increase of the volume of the sphere when the radius is 10 centimeters is (16\pi , cm^3/s).

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The rate of increase of the volume of the sphere can be calculated using the formula for the volume of a sphere and the given rate of increase of the radius.

Volume of a sphere formula: (V = \frac{4}{3} \pi r^3)

Taking the derivative of both sides with respect to time ((t)) gives:

[\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}]

Given:

- (\frac{dr}{dt} = 0.04) centimeters per second (the rate at which the radius is increasing)
- (r = 10) centimeters (the radius of the sphere at the given time)

Substitute the values into the formula:

[\frac{dV}{dt} = 4\pi (10)^2 \cdot 0.04]

Calculate:

[\frac{dV}{dt} = 4\pi \times 100 \times 0.04]

[\frac{dV}{dt} = 16\pi]

So, the rate of increase of the volume of the sphere when the radius is 10 centimeters is (16\pi) cubic centimeters 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.

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.

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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