an ice cube that is 8 cm on each side is melting at a rate of 4 cm^3 per min. how fast is the length of the side decreasing?
To find how fast the length of the side is decreasing, we need to find the derivative of the length of the side with respect to time.
Let's denote the length of the side as s and the rate at which the ice cube is melting as V.
We are given that the ice cube is melting at a rate of 4 cm^3 per minute, so we have dV/dt = 4 cm^3/min.
We are asked to find ds/dt, the rate at which the length of the side is decreasing.
We know that the volume V of a cube is given by V = s^3. We can differentiate both sides of this equation with respect to time t:
dV/dt = d/dt(s^3)
Since the volume V is changing with time, we can rewrite dV/dt as V' and ds/dt as s'.
V' = d/dt(s^3)
Using the chain rule, the derivative of s^3 with respect to t is:
V' = 3s^2 * ds/dt
We know that V' = 4 cm^3/min, so we can substitute this value into the equation:
4 cm^3/min = 3s^2 * ds/dt
Now we can solve for ds/dt:
ds/dt = 4 cm^3/min / (3s^2)
We are given that the length of the side s is 8 cm, so we can substitute this value into the equation:
ds/dt = 4 cm^3/min / (3 * (8 cm)^2)
Simplifying the equation further:
ds/dt = 4 cm^3/min / (3 * 64 cm^2)
ds/dt = 4 cm^3/min / (192 cm^2)
Therefore, the rate at which the length of the side is decreasing is 0.0208 cm/min (rounded to four decimal places).
To find out how fast the length of the side is decreasing, we need to differentiate the volume of the ice cube with respect to time.
Given:
Side length of the ice cube = 8 cm
Rate of melting = 4 cm³/min
The volume (V) of a cube is given by the formula:
V = s³
where s is the length of the side of the cube.
Differentiating both sides of the equation with respect to time (t), we get:
dV/dt = 3s²(ds/dt)
Here, dV/dt represents the rate of change of volume with respect to time (which is the rate of melting), and ds/dt represents the rate of change of the side length with respect to time (which is what we need to find).
Plugging in the given values:
4 cm³/min = 3(8 cm)²(ds/dt)
Simplifying the equation, we get:
4 cm³/min = 192 cm²(ds/dt)
Now, we can solve for ds/dt:
(ds/dt) = (4 cm³/min) / (192 cm²)
(ds/dt) = 0.0208 cm/min
Therefore, the length of the side is decreasing at a rate of approximately 0.0208 cm/min.
v = s^3
dv/dt = 3s^2 ds/dt
so, when s=8, v512 and we have
-4 = 192 ds/dt
now just solve for ds/dt