Question

An ice cube that is 6 cm on each side is melting at a rate of 5cm^3 per minute. How fast is the length of the side decreasing? At the moment when the ice cube is 6cm on each side, the side length is decreasing at a rate of approximately ________ cm per minute?

Answer 1

The length of the side of the ice cube is decreasing at a rate of approximately -0.0926 cm per minute.

Step-by-step explanation:

To find the rate at which the length of the side of the ice cube is decreasing, we need to find the derivative of the volume of the ice cube with respect to time. The volume of an ice cube with side length s is given by V = s^3. Taking the derivative with respect to time, we get dV/dt = 3s^2 ds/dt, where ds/dt is the rate at which the side length is changing.

Given that the ice cube is melting at a rate of 5 cm^3 per minute, we have dV/dt = -5. We also know that the length of the side of the ice cube is 6 cm, so s = 6. Substituting these values into the equation, we have -5 = 3(6^2)(ds/dt), which simplifies to ds/dt = -5/(3(6^2)), which is approximately -0.0926 cm per minute.

Answer 2

The length of the side is decreasing at a rate of approximately 0.0463 cm per minute.

Step-by-step explanation:

Let's denote the length of the side of the ice cube as x cm. Since the ice cube is melting at a rate of 5 cm^3 per minute, and the ice cube is a perfect cube, the volume of the ice cube can be expressed as V = x^3 cm^3. The rate at which the length of the side is decreasing can be found by differentiating both sides of the equation:

dV/dt = d/dt (x^3) = 3x^2 (dx/dt)

Given that dV/dt = 5 cm^3/min, we can substitute this into the equation and solve for dx/dt:

5 cm^3/min = 3x^2 (dx/dt)

Since the side length x is 6 cm, we can substitute this value into the equation and solve for dx/dt:

5 cm^3/min = 3(6^2) (dx/dt)

dx/dt = 5 cm^3/min ÷ 3(6^2) = 5 cm^3/min ÷ 108 = 0.0463 cm/min

Therefore, the length of the side is decreasing at a rate of approximately 0.0463 cm per minute when the ice cube is 6 cm on each side.