Question

Sodium chlorate crystals are easy to grow in the shape of cubes by allowing a solution of water and sodium chlorate to evaporate slowly. If V is the volume of such a cube with side length x, calculate the derivative when x = 3 mm and explain its meaning.

Answer 1

Answer:

• The derivative when x = 3 mm is 27.

         V'(3) = 27

• This means that the volume's rate of change when x = 3 (when the sides are 3mm) is 27.

Step-by-step explanation:

The volume of a cube with side length x can be expressed as:

$volume = (height)(width)(length)$

for a cube: ×$(height)=(width)=(length)$

-Since the side length is x then:

$V=x^{3}$

The derivative of V is:

$V'=3x^{2}$

-When x = 3 the derivative is:

$V'=3(3)^{2}$

-Therefore,

$V'(3)=27$

Keep in mind that a derivative represents the slope of the tangent line at a given point. It can also mean the rate of change at which the volume is changing when the side lenght is 3.