As an ice cube melts its surface area in decreasing at a rate of 6 cm^2 / sec. Find the rate at which the length of each side is decreasing at the moment when each side has length 2 cm.
cube side = x, so the area
a = 6x^2
da/dt = 12s dx/dt
Now just plug in your numbers.
To find the rate at which the length of each side is decreasing, we need to use related rates. Let's denote the length of each side of the ice cube as "s" (in cm) and the surface area as "A" (in cm²).
Given: dA/dt = -6 cm²/sec (the rate at which the surface area is decreasing)
We need to find ds/dt when s = 2 cm.
The surface area of a cube is given by:
A = 6s².
Now, differentiate both sides of the equation with respect to time (t):
dA/dt = d(6s²)/dt.
To find the derivative of 6s², we use the chain rule:
dA/dt = 12s(ds/dt).
We know that dA/dt = -6 cm²/sec and s = 2 cm. Plugging in these values, we get:
-6 = 12(2)(ds/dt).
Simplifying the equation:
-6 = 24(ds/dt).
Now, solve for ds/dt:
ds/dt = -6/24.
Simplifying further:
ds/dt = -1/4.
Therefore, the rate at which the length of each side is decreasing when each side has a length of 2 cm is -1/4 cm/sec.