All wedges of a cube are expanding at a rate of 6 cm/sec .How fast is the surface area changing when each edge is (a) 2 cm and (b) 10cm
To find how fast the surface area is changing, we need to differentiate the surface area formula with respect to time and then substitute the given values.
The surface area of a cube is given by the formula
A = 6s^2,
where A represents the surface area and s represents the length of each edge.
Let's differentiate both sides of the equation with respect to time (t):
dA/dt = d/dt(6s^2).
Using the chain rule, we know that:
dA/dt = 12s * ds/dt.
Now, we can substitute the given values for ds/dt when s = 2cm and s = 10cm.
(a) When s = 2cm:
dA/dt = 12s * ds/dt
dA/dt = 12(2) * 6 cm/sec
dA/dt = 144 cm^2/sec.
So, when each edge is 2 cm, the surface area is changing at a rate of 144 cm^2/sec.
(b) When s = 10cm:
dA/dt = 12s * ds/dt
dA/dt = 12(10) * 6 cm/sec
dA/dt = 720 cm^2/sec.
So, when each edge is 10 cm, the surface area is changing at a rate of 720 cm^2/sec.