The volume of a cube increases at a constant rate of 10 cm^3 per second. find the rate of change in its total surface area at the instant when its sides are 20 cm long.
side of cube ---- x cm
volume = x^3
d(volume)/dt = 3x^2 dx/dt
when c = 20, d(volume)/dt = 10 cm^3/sec
10 = 3(20^2) dx/dt
dx/dt = 10/1200 = 1/120 cm/sec
Surface area = SA = 6x^2
d(SA)/dt = 12x dx/dt
so when x = 20, as above
d(SA)/dt = 12(20)(1/120) cm^2/sec
= 2 cm^2/sec
Well, well, well, we've got some mathematical fun coming our way! Okay, let's dive right in.
We can start by finding the rate of change of the side length of the cube. Since the volume of a cube is given by V = s^3 (where s represents the length of the sides), we can differentiate both sides with respect to time (t) and get dV/dt = 3s^2(ds/dt).
We know that dV/dt = 10 cm^3/s (the volume is increasing at a constant rate), and we want to find ds/dt when s = 20 cm. So, plug in what we know and solve for ds/dt:
10 cm^3/s = 3(20)^2(ds/dt)
10 = 1200(ds/dt)
ds/dt = 10/1200
ds/dt = 1/120 cm/s.
My funny bone tells me that the rate of change in the total surface area of a cube is given by dA/dt = 6s(ds/dt) (where A represents the total surface area). Plugging in what we know, we get:
dA/dt = 6(20)(1/120)
dA/dt = 1/2 cm^2/s.
Voila! The rate of change in the total surface area of the cube, at the instant when its sides are 20 cm long, is 1/2 cm^2/s. Keep up the great math work, my friend!
To find the rate of change in the total surface area of a cube, we need to know the relationship between the volume and the surface area of a cube.
The volume (V) of a cube with side length (s) is given by the formula V = s^3.
The total surface area (A) of a cube with side length (s) is given by the formula A = 6s^2.
In this case, the volume is increasing at a constant rate of 10 cm^3 per second. Therefore, the rate of change of volume (dV/dt) is 10 cm^3/s.
We are asked to find the rate of change of the surface area (dA/dt) at the instant when the sides of the cube are 20 cm long.
Step 1: Find the rate of change of side length (ds/dt).
We know that the volume of the cube is given by V = s^3. Differentiating both sides of the equation with respect to time, we get:
dV/dt = d(s^3)/dt
10 cm^3/s = 3s^2 * ds/dt
Step 2: Find the rate of change of surface area (dA/dt).
We know that the surface area of the cube is given by A = 6s^2. Differentiating both sides of the equation with respect to time, we get:
dA/dt = d(6s^2)/dt
dA/dt = 6 * 2s * ds/dt
dA/dt = 12s * ds/dt
Step 3: Find the rate of change of surface area at the instant when the sides of the cube are 20 cm long.
Since we are given that the sides of the cube are 20 cm long, we substitute s = 20 cm into the equation for dA/dt:
dA/dt = 12 * 20 cm * ds/dt
Step 4: Substitute the rate of change of volume (dV/dt) into the equation for ds/dt.
From Step 1, we found that 10 cm^3/s = 3s^2 * ds/dt. Since the sides of the cube are 20 cm long, we substitute s = 20 cm into the equation:
10 cm^3/s = 3 * (20 cm)^2 * ds/dt
10 cm^3/s = 3 * 400 cm^2 * ds/dt
ds/dt = 10 cm^3/s / (3 * 400 cm^2)
ds/dt = 10 cm/s / (3 * 400 cm)
ds/dt = 10 / (3 * 400) cm/s
Step 5: Substitute the value of ds/dt into the equation for dA/dt.
Using the value of ds/dt from Step 4, substitute into the equation for dA/dt:
dA/dt = 12 * 20 cm * ds/dt
dA/dt = 12 * 20 cm * (10 / (3 * 400) cm/s)
dA/dt = (12 * 20 * 10) / (3 * 400) cm^2/s
dA/dt = 24/3 cm^2/s
dA/dt = 8 cm^2/s
So, the rate of change in the total surface area of the cube at the instant when its sides are 20 cm long is 8 cm^2/s.
To find the rate of change in the total surface area of the cube, we need to differentiate the formula for the total surface area of a cube with respect to time.
The total surface area (A) of a cube is given by the formula: A = 6s^2, where s is the length of each side of the cube.
Since we know that the volume (V) of the cube is increasing at a constant rate of 10 cm^3 per second, we can express the volume as a function of time (t): V = 10t.
We need to find an expression for the side length (s) in terms of time (t). The formula for the volume of a cube is given by V = s^3. Rearranging this formula, we get s = V^(1/3).
Substituting the expression for volume, we have: s = (10t)^(1/3).
Now, we can substitute this expression for s into the formula for the total surface area: A = 6s^2 = 6[(10t)^(1/3)]^2 = 6 * 10^(2/3) * t^(2/3).
To find the rate of change in the total surface area (dA/dt) with respect to time, we differentiate A with respect to t:
dA/dt = d/dt[6 * 10^(2/3) * t^(2/3)].
Using the power rule of differentiation, we get:
dA/dt = 6 * 10^(2/3) * (2/3) * t^(-1/3).
Now we have the expression for the rate of change in the total surface area of the cube with respect to time. To find the rate of change at the instant when the sides are 20 cm long, substitute t = 20 into the expression:
dA/dt = 6 * 10^(2/3) * (2/3) * (20)^(-1/3).
Calculating this expression will give you the rate of change in the total surface area of the cube at that instant.