The edges of a cube increase at a rate of 2 cm/s. How fast is the volume changing when the length of each edge is 50 cm?

Verified step by step guidance

1. First, identify the formula for the volume of a cube, which is V = a^3, where 'a' is the length of an edge.
2. Next, differentiate the volume formula with respect to time 't' to find the rate of change of the volume. This gives us dV/dt = 3a^2 * (da/dt).
3. Substitute the given rate of change of the edge length, da/dt = 2 cm/s, into the differentiated formula.
4. Substitute the given edge length, a = 50 cm, into the formula. This will allow you to calculate the rate at which the volume is changing at that specific edge length.
5. Finally, compute the expression to find the rate of change of the volume, dV/dt, using the substituted values.

Key Concepts

Related Rates

Related rates involve finding the rate at which one quantity changes in relation to another. In this problem, we need to determine how the volume of the cube changes as the length of its edges increases. This requires applying the chain rule of differentiation to relate the rates of change of the edge length and the volume.

Volume of a Cube

The volume of a cube is calculated using the formula V = s^3, where s is the length of an edge. Understanding this formula is crucial because it allows us to express the volume in terms of the edge length, enabling us to differentiate it with respect to time to find the rate of change of volume.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. In this context, we differentiate the volume function with respect to time to find how fast the volume is changing as the edge length changes. This process involves applying the power rule and the chain rule to relate the rates of change.