the side of a cube is expanding at a constant rate of 2 centimeters per sesond. what is the instantaneous rate of change of the surface area of the cube, in cm2 per second, when its volume is 27 cubic cm
For a cube, SA = 6s^2. Differentiate with respect to time.
dSA/dt = 12s(ds/dt)
ds/dt is given as 2cm/s.
When V = 27cm^3, s = 3cm. dSA/dt=?
72
Well, if the side of the cube is expanding at a constant rate of 2 centimeters per second, it means that the rate of change of the side length is 2 cm/s.
Now, let's denote the side length of the cube as "s." Since the volume of a cube is given by V = s^3, and we know that the volume is 27 cubic cm, we can solve for the side length:
27 = s^3
Take the cubic root of both sides to find s:
s = 3 cm
So, when the volume is 27 cubic cm, the side length of the cube is 3 cm.
Now, let's find the instantaneous rate of change of the surface area of the cube.
The surface area of a cube is given by A = 6s^2, where s is the side length. Differentiating both sides of this equation with respect to time (t), we get:
dA/dt = 12s * ds/dt
Now, we know that ds/dt (the rate of change of the side length) is 2 cm/s.
Plugging in the values, we have:
dA/dt = 12(3)(2)
dA/dt = 72 cm^2/s
Therefore, when the volume is 27 cubic cm, the instantaneous rate of change of the surface area of the cube is 72 cm^2/s.
To find the instantaneous rate of change of the surface area of a cube, we need to use derivatives. Let's proceed step-by-step:
Step 1: Recall the formulas for the surface area and volume of a cube.
- The surface area of a cube is given by the equation: S = 6s^2, where s is the length of one side of the cube.
- The volume of a cube is given by the equation: V = s^3.
Step 2: Determine the expression for the surface area of the cube in terms of its volume.
Since we know the volume is 27 cubic cm, we can rearrange the formula for volume to solve for s: s = V^(1/3), where V is the volume.
Substituting the volume value of 27 into the equation, we get s = 27^(1/3) = 3 cm.
Step 3: Calculate the rate of change of the surface area with respect to time.
Since the side length of the cube is expanding at a constant rate of 2 cm/s, the rate of change of the side length (ds/dt) is 2 cm/s.
To find the rate of change of the surface area (dS/dt) with respect to time, we need to take the derivative of the surface area equation with respect to the side length: dS/dt = d(6s^2)/dt = 12s(ds/dt).
Step 4: Substitute the known values into the formula.
Substituting s = 3 cm and ds/dt = 2 cm/s into dS/dt = 12s(ds/dt), we get:
dS/dt = 12 * 3 * 2 = 72 cm^2/s.
Therefore, the instantaneous rate of change of the surface area of the cube, when its volume is 27 cubic cm, is 72 cm^2 per second.
To find the instantaneous rate of change of the surface area of the cube, we need to use calculus. The formula for the surface area of a cube is given by A = 6s^2, where A is the surface area and s is the length of the side of the cube.
Given that the side of the cube is expanding at a constant rate of 2 centimeters per second, we can determine the rate of change of the side length with respect to time.
Let's define s(t) as the length of the side of the cube at time t. Since the side length is increasing at a rate of 2 centimeters per second, the derivative of s(t) with respect to time will be ds/dt = 2.
Now, we need to find the derivative of the surface area with respect to time to determine the instantaneous rate of change. We can differentiate the surface area formula with respect to the side length s:
dA/dt = d/dt(6s^2)
To find dA/dt, we will differentiate each term with respect to t:
dA/dt = 12s * ds/dt
Substituting ds/dt = 2 and s = 3 (since volume is given as 27 cubic cm, and the volume of a cube is V = s^3), we can calculate the instantaneous rate of change:
dA/dt = 12 * 3 * 2
dA/dt = 72 cm^2 per second
Therefore, the instantaneous rate of change of the surface area of the cube, when its volume is 27 cubic cm, is 72 cm^2 per second.