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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.

Respuesta :

creamymoorfoot

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:

[tex]volume = (height)(width)(length)[/tex]

for a cube: ×[tex](height)=(width)=(length)[/tex]

-Since the side length is x then:

[tex]V=x^{3}[/tex]

The derivative of V is:

[tex]V'=3x^{2}[/tex]

-When x = 3 the derivative is:

[tex]V'=3(3)^{2}[/tex]

-Therefore,

[tex]V'(3)=27[/tex]

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.

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