An ice cube is 3 by 3 by 3 inches is melting in such a way that the length of one of its side is decreasing at a rate of half an inch per minute. Find the rate at which its surface area is decreasing at the moment when the volume of the cube is 8 cubic inches
v = s^3
dv/dt = 3s^2 ds/dt
now just plug in your numbers.
s.a. = 6 s²
da/dt = 12 s ds/dt
v = s³ ... 8 = 2³
To find the rate at which the surface area of the melting ice cube is decreasing, we need to understand the relationships between the dimensions: side length, surface area, and volume.
Let's start by defining some variables:
Let \(x\) be the length of one side of the ice cube at any given time (in inches).
Let \(A\) be the surface area of the ice cube at that time (in square inches).
Let \(V\) be the volume of the ice cube at that time (in cubic inches).
We know that the side length is decreasing at a rate of half an inch per minute. Therefore, the rate of change of the side length (\(\frac{dx}{dt}\)) is -0.5 inches per minute.
The surface area of a cube with side length \(x\) can be calculated using the formula: \(A = 6x^2\).
So the rate of change of the surface area (\(\frac{dA}{dt}\)) can be found by differentiating this equation with respect to time.
\(\frac{dA}{dt} = \frac{d}{dt}(6x^2)\)
\(\frac{dA}{dt} = 12x\frac{dx}{dt}\)
We need to find the rate of change of the surface area at the time when the volume of the cube is 8 cubic inches, indicating that \(V = 8\).
The volume of a cube with side length \(x\) can be calculated using the formula: \(V = x^3\).
So we have: \(8 = x^3\).
To solve for \(x\), we take the cube root of both sides:
\(x = \sqrt[3]{8}\)
\(x = 2\) inches
Now that we know the value of \(x\), we can substitute it into the equation we derived earlier for \(\frac{dA}{dt}\):
\(\frac{dA}{dt} = 12x\frac{dx}{dt}\)
\(\frac{dA}{dt} = 12(2)(-0.5)\)
\(\frac{dA}{dt} = -12\) square inches per minute
Therefore, the rate at which the surface area of the melting ice cube is decreasing at the moment when the volume of the cube is 8 cubic inches is -12 square inches per minute.