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?
http://www.jiskha.com/display.cgi?id=1293829458
To find the rate at which the volume of the cubical crystal is increasing, we can use the formula for the volume of a cube. The volume of a cube is given by V = s^3, where s is the length of each side of the cube.
Given that the edges of the cube are growing at a rate of 2 mm per hour, we can find the rate at which the volume is increasing by taking the derivative with respect to time (t).
Let's find the derivative of the volume function V = s^3 with respect to time:
dV/dt = 3s^2 * ds/dt
Now, plug in the values given in the problem:
s = 5 mm (length of each side of the cube)
ds/dt = 2 mm/hour (rate of change of the side length)
Substituting these values into the derivative expression, we get:
dV/dt = 3(5 mm)^2 * 2 mm/hour
Simplifying the expression:
dV/dt = 3 * 25 mm^2 * 2 mm/hour
dV/dt = 150 mm^3/hour
Therefore, the volume of the cubical crystal is increasing at a rate of 150 mm^3 per hour.