# If a snowball melts so that its surface area decreases at a rate of 3 cm2/min, how do you find the rate at which the diameter decreases when the diameter is 12 cm?

diameter is decreasing at rate of

By the chain rule we have:

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To find the rate at which the diameter decreases when the diameter is 12 cm, you can use the formula for the surface area of a sphere:

[ A = 4\pi r^2 ]

Differentiating both sides with respect to time ( t ), you get:

[ \frac{dA}{dt} = 8\pi r \frac{dr}{dt} ]

Given that ( \frac{dA}{dt} = -3 , \text{cm}^2/\text{min} ) (negative because the surface area is decreasing), and when ( r = 6 , \text{cm} ), you can substitute these values into the equation and solve for ( \frac{dr}{dt} ).

[ -3 = 8\pi (6) \frac{dr}{dt} ]

Solving for ( \frac{dr}{dt} ):

[ \frac{dr}{dt} = -\frac{1}{8\pi} , \text{cm/min} ]

So, the rate at which the diameter decreases when the diameter is 12 cm is ( -\frac{1}{8\pi} , \text{cm/min} ).

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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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