An ice cube that is 3 cm on each side is melting at a rate of 2 cm cubed per minute. how fast is the length of the side decreasing?
given: dV/dt = -2 cm^3/min
V= x^3
dV/dt = 3x^2 dx/dt
-2 = 3(9)dx/dt
dx/dt = -2/27 cm/min
decreasing at 2/27 cm/min
Ah, the cold, chilling dilemma of the melting ice cube! Let's put on our thinking caps and do some math.
The volume of the ice cube is given as 3 cm³, and it's melting at a rate of 2 cm³ per minute. So, using our trusty Time Lord skills, we can say that after one minute, the volume will be reduced to 1 cm³.
Now, since the ice cube is a perfect cube, all sides have the same length. Let's call the length of each side "s." We want to find how fast the length of the side is decreasing, or ds/dt.
Since the volume of a cube is given by V = s³, we can differentiate both sides with respect to time to get dV/dt = 3s²(ds/dt).
We know that dV/dt = -2, as the volume is decreasing. So, we can plug in these values and solve for ds/dt:
-2 = 3s²(ds/dt)
ds/dt = -2/(3s²)
So, the length of the side is decreasing at a rate of -2/(3s²) cm per minute. Just be careful, you wouldn't want to be around to witness an ice cube's identity crisis!
To find the rate at which the length of the side is decreasing, we need to calculate the derivative of the side length with respect to time.
Let's assume that the length of the side of the ice cube is represented by the variable "s" (cm), and the time is represented by the variable "t" (minutes).
Given:
Side length of the ice cube, s = 3 cm
Rate of melting, V = 2 cm^3/min
We know that the volume of a cube is given by V = s^3.
Differentiating both sides of the equation with respect to time t, we get:
dV/dt = d(s^3)/dt
Using the chain rule, the derivative of s^3 with respect to t is:
d(s^3)/dt = 3s^2 * ds/dt
Rearranging the equation to solve for ds/dt (the rate at which the length of the side is decreasing), we have:
ds/dt = (dV/dt) / (3s^2)
Substituting the given values, we get:
ds/dt = (2 cm^3/min) / (3 * (3 cm)^2)
= 2 cm^3/min / (3 * 9 cm^2)
= 2 cm^3/min / 27 cm^2
= 2/27 cm/min
Therefore, the length of the side is decreasing at a rate of 2/27 cm per minute.
To determine how fast the length of the side of the ice cube is decreasing, we can use the concept of related rates.
Let's denote the length of the side of the ice cube as "s" and the rate at which it is decreasing as "ds/dt."
Given that the volume of the ice cube is decreasing at a rate of 2 cm³/min, we can express this as dV/dt = -2 cm³/min, where dV/dt represents the rate of change of the volume.
The volume of a cube is given by V = s³, and we can differentiate both sides of this equation with respect to time (t) to find the related rate:
dV/dt = d/dt (s³)
Since the volume of the ice cube is decreasing, dV/dt is negative, so we have:
-2 = 3s²(ds/dt)
Now, we can solve this equation for ds/dt, the rate at which the length of the side of the ice cube is decreasing:
ds/dt = -(2)/(3s²)
Substituting the given side length, s = 3 cm, into the equation:
ds/dt = -(2)/(3(3)²)
= -(2)/(27)
Therefore, the rate at which the length of the side of the ice cube is decreasing is -(2/27) cm/min.