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?

Answer 1

The rate of increase would be #16pi#

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

#(dr)/dt = 0.04#
The volume of a sphere is #4/3pir^3# Differentiate this to find the rate of the change of volume:
#4pir^2(dr)/dt# Plug in our variables: #4pi(10)^2(0.04)# This can evaluate to equal: #16pi#

Hope this helps!

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Answer 2

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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Answer 3

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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Answer from HIX Tutor

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.

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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