Razón de cambio - Problema 1

Summary

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Takeaways

• 😀 The problem involves finding the rate of change of the volume of a cube as its edge length increases over time.
• 😀 The edge length of the cube is growing at a rate of 2 cm per second.
• 😀 The volume of a cube is calculated using the formula V = a³, where 'a' is the edge length.
• 😀 To find the rate at which the volume changes, we differentiate the volume formula with respect to time.
• 😀 The derivative of the volume formula is dV/dt = 3a² * (da/dt), where da/dt is the rate of change of the edge length.
• 😀 The specific problem asks for the rate of change of the volume when the edge length is 5 cm.
• 😀 The rate of change of the edge length is given as da/dt = 2 cm per second.
• 😀 By substituting the values into the derivative formula, we calculate dV/dt = 3 * (5)² * 2 = 150 cm³/s.
• 😀 The rate at which the volume of the cube is changing at the moment when the edge length is 5 cm is 150 cubic centimeters per second.
• 😀 This problem is a practical example of related rates, showing how derivatives can be used to find rates of change in geometric contexts.

Q & A

What is the formula for the volume of a cube?

The volume of a cube is given by the formula V = a³, where 'a' is the length of one edge of the cube.

What is the given rate of change of the edge length in the problem?

The rate of change of the edge length is given as 2 cm per second.

What is the edge length of the cube at the specific moment the rate of change is being calculated?

The edge length of the cube at that moment is 5 cm.

What mathematical operation is used to find the rate of change of the volume with respect to time?

To find the rate of change of the volume, we differentiate the volume formula with respect to time.

What is the derivative of the volume formula with respect to time?

The derivative of the volume formula V = a³ with respect to time is dV/dt = 3a² * (da/dt), where da/dt is the rate of change of the edge length.

What does the expression dV/dt represent in the context of this problem?

The expression dV/dt represents the rate of change of the volume of the cube with respect to time.

How do we interpret the value of da/dt in this problem?

The value of da/dt is the rate at which the edge length of the cube is changing, and in this case, it is given as 2 cm per second.

What is the significance of the value '5 cm' in the calculation?

The value '5 cm' is the current length of the edge of the cube when the rate of change of the volume is being calculated.

What is the final result for the rate of change of the volume?

The rate of change of the volume is 150 cubic centimeters per second (150 cm³/s).

Why is the rate of change of volume expressed in cubic centimeters per second?

The rate of change of volume is expressed in cubic centimeters per second because the volume is measured in cubic centimeters, and the time is measured in seconds.