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

Let c=length of each side,

Area of cube, A = 6c²
Rate of change of area with respect to side = dA/dc = 12c
Rate of change of Area can be found by the chain rule,

dA/dt
= dA/dc * dc/dt
= 12c * 6 cm/s
=72c cm/s ...(1)
(a) c=2 cm,
(b) c=10 cm
Substitute values of c in expression (1) to get the rate of change of area with respect to time.

To find how fast the surface area is changing, we need to use the formula for the surface area of a cube, which is:

SA = 6a^2

where SA is the surface area and a is the length of each edge.

To find how fast the surface area is changing, we can take the derivative of this equation with respect to time (t), which gives us:

dSA/dt = 12a(da/dt)

where dSA/dt is the rate of change of the surface area, da/dt is the rate of change of the edge length, and a is the current edge length.

For part (a), the edge length is 2 cm, and the rate of change of the edge length (da/dt) is given as 6 cm/sec. Substituting these values into the equation, we get:

dSA/dt = 12(2)(6) = 144 cm^2/sec

Therefore, the surface area is changing at a rate of 144 cm^2/sec when each edge is 2 cm.

For part (b), the edge length is 10 cm, and the rate of change of the edge length (da/dt) is still given as 6 cm/sec. Substituting these values into the equation, we get:

dSA/dt = 12(10)(6) = 720 cm^2/sec

Therefore, the surface area is changing at a rate of 720 cm^2/sec when each edge is 10 cm.

To find how fast the surface area of the cube is changing, we can use the formula for the surface area of a cube:

Surface Area = 6 * (edge length)^2

Let's calculate the rate of change of the surface area when each edge is 2 cm:

Given: d(edge length)/dt = 6 cm/sec
We need to find: d(Surface Area)/dt

Using the chain rule of differentiation, we have:

d(Surface Area)/dt = d(Surface Area)/d(edge length) * d(edge length)/dt

Differentiating the surface area formula with respect to the edge length gives us:

d(Surface Area)/d(edge length) = 12 * (edge length)

Now we substitute the given values:

d(edge length)/dt = 6 cm/sec
edge length = 2 cm

Plugging these values into the equation, we get:

d(Surface Area)/dt = 12 * (2) * (6)
= 144 cm^2/sec

Therefore, when each edge is 2 cm, the surface area of the cube is changing at a rate of 144 cm^2/sec.

Now let's calculate the rate of change of the surface area when each edge is 10 cm:

Given: d(edge length)/dt = 6 cm/sec
We need to find: d(Surface Area)/dt

Using the same process, we differentiate the surface area formula:

d(Surface Area)/d(edge length) = 12 * (edge length)

Now we substitute the given values:

d(edge length)/dt = 6 cm/sec
edge length = 10 cm

Plugging these values into the equation, we get:

d(Surface Area)/dt = 12 * (10) * (6)
= 720 cm^2/sec

Therefore, when each edge is 10 cm, the surface area of the cube is changing at a rate of 720 cm^2/sec.