# A hypothetical cube shrinks at a rate of 8 m³/min. At what rate are the sides of the cube changing when the sides are 3 m each?

When the sides are

Identify the Rates of Change

We are asked to find the rate at which the sides are changing, so we want to

Find an Equation Relating the Variables

The volume of a cube is given by the equation

Differentiate To find the equation relating the variables and their rates of change.

Plug in what you know and solve for what you're looking for.

Answer the question

If you prefer to use units all the way through:

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To find the rate at which the sides of the cube are changing when each side is 3 meters long, we can use the formula for the volume of a cube, (V = s^3), where (V) is the volume and (s) is the length of a side.

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

(\frac{{dV}}{{dt}} = 3s^2 \frac{{ds}}{{dt}})

Given that ( \frac{{dV}}{{dt}} = -8 , m^3/min) and (s = 3 , m), we can solve for (\frac{{ds}}{{dt}}):

(-8 = 3(3^2) \frac{{ds}}{{dt}})

(-8 = 27 \frac{{ds}}{{dt}})

(\frac{{ds}}{{dt}} = \frac{{-8}}{{27}} , m/min)

So, when the sides of the cube are each 3 meters long, they are decreasing at a rate of (\frac{{-8}}{{27}}) meters per minute.

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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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