All edges of a cube are expanding at a rate of 8 centimeters per second.
(a) How fast is the volume changing when each edge is 4 cm?
(b) How fast is the volume changing when each edge is 10 cm?
Kind of confused...
V = x^3
dx/dt = 8
dV/dt = 3x^2 dx/dt
(a) When x = 4, ???
(b) When x = 10, ???
256
To find the rate of change of the volume, we need to differentiate the volume formula V = x^3 with respect to time. The given rate of change of the edges is dx/dt = 8 cm/s.
Let's calculate the rate of change of the volume at each given edge length.
(a) When x = 4 cm:
Substituting x = 4 cm into the volume formula, we have V = (4)^3 = 64 cm³.
Differentiating the volume formula with respect to time, we have dV/dt = 3(4)^2(dx/dt) = 3(16)(8).
So, when each edge is 4 cm, the volume is changing at a rate of dV/dt = 384 cm³/s.
(b) When x = 10 cm:
Substituting x = 10 cm into the volume formula, we have V = (10)^3 = 1000 cm³.
Differentiating the volume formula with respect to time, we have dV/dt = 3(10)^2(dx/dt) = 3(100)(8).
So, when each edge is 10 cm, the volume is changing at a rate of dV/dt = 2400 cm³/s.
Therefore, when each edge is 4 cm, the volume is changing at a rate of 384 cm³/s, and when each edge is 10 cm, the volume is changing at a rate of 2400 cm³/s.
To find how fast the volume of the cube is changing, we can use the formula:
dV/dt = 3x^2 dx/dt
where V is the volume, t is time, x is the length of the edge of the cube, and dx/dt is the rate at which the edge is expanding.
Given that dx/dt = 8 centimeters per second, we can substitute this value in the formula:
dV/dt = 3x^2 (8)
Now, let's solve for the values requested:
(a) When x = 4 cm:
Substituting x = 4 cm into the formula:
dV/dt = 3(4^2) (8)
= 3(16) (8)
= 48 (8)
= 384 cm^3/s
Therefore, when each edge is 4 cm, the volume is changing at a rate of 384 cubic centimeters per second.
(b) When x = 10 cm:
Substituting x = 10 cm into the formula:
dV/dt = 3(10^2) (8)
= 3(100) (8)
= 300 (8)
= 2400 cm^3/s
Therefore, when each edge is 10 cm, the volume is changing at a rate of 2400 cubic centimeters per second.
I hope this clarifies it for you! Let me know if you have any further questions.
a) dVdt=3x^2 dx/dt
=2(16)(8) cm^3/sec