At a certain point in time, a cubical crystal has edges 5 mm in length, and these edges are growing at a rate of 2 mm per hour. How fast is the volume increasing?
V = x^3
dV/dx = 3 x^2
dV/dt = DV/dx*dx/dt
enough?
yeah thanks :)
Good. You are welcome.
To find the rate at which the volume of the cubical crystal is increasing, we need to differentiate the volume with respect to time.
Let's denote the length of an edge of the cube as "x" and the volume of the cube as "V". According to the given information, the length of an edge is changing at a rate of 2 mm per hour, which means dx/dt = 2 mm/hr.
The volume of a cube is given by V = x^3. To find the rate at which the volume is increasing, we need to differentiate the volume function with respect to time:
dV/dt = d/dt (x^3)
To differentiate x^3 with respect to time, we can apply the chain rule:
dV/dt = 3x^2 * dx/dt
Substituting the given values, we have:
dx/dt = 2 mm/hr
x = 5 mm
Now, we can calculate the rate of volume increase:
dV/dt = 3x^2 * dx/dt
= 3(5^2) * 2
= 3 * 25 * 2
= 150 mm^3/hr
Therefore, the volume of the cubical crystal is increasing at a rate of 150 cubic millimeters per hour.