the edge of a cube is increasing at the rate of .05 centimeters per second. in terms of the side of the cube s, what is the rate of change of the volume of the cube, in cm^3/sec
V = s^3
dV/dt = 3s^2 ds/dt
= 3s^2(.05)
= .15 s^2
I don't think your question was complete.
It probably said "what is the rate of change of the volume of the cube, in cm^3/sec, when s = ...."
Sub in the value of the given s into the dV/dt,
otherwise the above answer stands as is.
Well, well, well! We have a little cube here that's growing, huh? How exciting! Now, let's get down to business.
The volume of a cube is given by the formula V = s^3, where s is the side length of the cube.
Since the edge of the cube is increasing at a rate of 0.05 cm per second, that means the side length s is also increasing at a rate of 0.05 cm per second.
Now, to find the rate of change of the volume, we need to differentiate the volume formula with respect to time.
dV/dt = 3s^2 * ds/dt
Here, dV/dt represents the rate of change of volume, and ds/dt represents the rate of change of the side length.
Substituting the given values, we get:
dV/dt = 3s^2 * (0.05)
So, the rate of change of the volume of the cube is 0.15s^2 cm^3/sec.
Now, go forth and impress your friends with your newfound knowledge of growing cubes!
To find the rate of change of the volume of a cube with respect to time, we can use the formula for the volume of a cube:
V = s^3
Where:
V = Volume of the cube
s = Side length of the cube
The rate of change of the volume, dV/dt, can be found using the chain rule. But first, let's express V in terms of t:
V = (s(t))^3
Here, s(t) represents the side length of the cube as a function of time. Given that the side length is increasing at a rate of 0.05 cm/s, we can rewrite s(t) as:
s(t) = s + 0.05t
Now, we can substitute s(t) into the formula for V:
V = (s + 0.05t)^3
Next, let's differentiate V with respect to time using the chain rule:
dV/dt = 3(s + 0.05t)^2 * (0 + 0.05)
Since we want to find the rate of change of the volume, we can simplify this expression:
dV/dt = 0.15(s + 0.05t)^2
Therefore, the rate of change of the volume of the cube is 0.15(s + 0.05t)^2 cm^3/sec.
To find the rate of change of the volume of the cube, we need to differentiate the volume function with respect to time.
The volume of a cube is given by V = s^3, where s is the length of one side of the cube.
Differentiating both sides of the equation with respect to time (t), we have:
dV/dt = 3s^2 * ds/dt
where dV/dt is the rate of change of volume, ds/dt is the rate of change of the side length, and s is the side length of the cube.
Given that ds/dt = 0.05 cm/sec (rate of change of the side length), we can substitute it into the equation:
dV/dt = 3s^2 * (0.05)
Now, let's say we know the length of one side of the cube, s. Once we substitute it into the equation, we will be able to calculate the rate of change of the volume of the cube in cm^3/sec.