a crystal in the shape of a cube is growing in a labatory. estimate the rate at which the surface area is changing with respect to the side length when the side length of the crystal is 3cm.
To estimate the rate at which the surface area is changing with respect to the side length of a crystal cube, we can use calculus.
The surface area, S, of a cube is given by S = 6s², where s is the side length of the cube.
To find the rate of change of S with respect to s, we need to take the derivative of the surface area function with respect to s, dS/ds.
In this case, we want to estimate this rate when the side length of the crystal cube is 3 cm, so we need to evaluate dS/ds at s = 3.
To find this derivative, we differentiate the surface area function with respect to s:
dS/ds = d/ds (6s²)
= 12s
Now, we can evaluate dS/ds at the given side length, s = 3:
dS/ds = 12(3)
= 36 cm²/cm
Therefore, when the side length of the crystal cube is 3 cm, the rate at which the surface area is changing with respect to the side length is 36 cm²/cm.