If each edge of a cube is increasing at the constant rate of 3 cm per sec how fast is the volume increasing when X, the length of an edge, is 10 cm long?
v = x^3
dv/dt = 3x^2 dx/dt
You have dx/dt and x, so figure dv/dt
To find the rate at which the volume of the cube is changing with respect to time, we can use the formula for the volume of a cube:
V = X^3
Where V is the volume and X is the length of an edge.
Differentiating both sides with respect to time t:
dV/dt = d(X^3)/dt
Using the power rule for differentiation:
dV/dt = 3X^2 * dX/dt
Given that dX/dt (the rate at which the length of an edge is changing) is constant at 3 cm/sec, and X = 10 cm:
dV/dt = 3(10)^2 * 3
Simplifying:
dV/dt = 300 cm^2/sec
Therefore, the volume of the cube is increasing at a rate of 300 cm^2/sec when the length of an edge is 10 cm.
To find how fast the volume of the cube is increasing, we can differentiate the volume formula with respect to time.
Let's denote the length of an edge of the cube as X and the volume as V.
The volume of a cube is given by V = X^3.
Now, we can differentiate both sides of the equation with respect to time (t), as we are interested in finding the rate of change of volume over time:
dV/dt = d/dt (X^3)
To differentiate X^3 with respect to t, we first need to apply the chain rule. The chain rule states that the derivative with respect to t of a function of another variable can be found by multiplying the derivative of the function with respect to the other variable by the derivative of the other variable with respect to t.
So, let's apply the chain rule to differentiate X^3 with respect to t:
dV/dt = d/dX (X^3) * dX/dt
The derivative of X^3 with respect to X can be found using the power rule of differentiation. According to the power rule, differentiate a variable raised to a constant power by multiplying the constant power by the variable raised to the power decreased by 1:
d/dX (X^3) = 3X^2
Now, let's differentiate X with respect to t. In the problem, we are told that "each edge of the cube is increasing at the constant rate of 3 cm per sec." So, the rate of change of X with respect to t (dX/dt) is equal to 3 cm per sec.
Substituting these values back into the equation:
dV/dt = (3X^2) * (3)
Now, we need to find the value of dV/dt when X is 10 cm long.
Substitute X = 10 into the expression:
dV/dt = (3 * 10^2) * (3)
Simplifying further:
dV/dt = 300 * 3 = 900 cm^3/sec
Therefore, the volume of the cube is increasing at a rate of 900 cm^3/sec when the length of an edge is 10 cm long.