To find: The value of at value of , if V is the volume of a cube with side length , and explain its meaning.
The value of is .
Calculate the value of when is and explain its meaning.
Use the below expression for volume.
Differentiate the volume equation.
Substitute the value for in the above equation.
The above value of is the rate at which the volume is increasing with the variable .
Therefore, is .
To show: The rate of change of the volume of a cube with respect to its edge length is equal to half the surface area of the cube and explain geometrically why this result is true.
Each cube will have 6 sides, and area of one side of the cube is .
Therefore, the surface area of the cube is as below.
Refer the equation (1).
Simplify the equation as below.
Substitute for .
Thus, the rate of change of the volume of the cube is approximately half of its surface area.
Assume the condition in which a cube’s side length is x and the change in length of side is on all sides.
Express the volume change, in terms of increased amount of length of side .
The value of is smaller and negligible.
Rewrite the above equation as below.
Hence, the result is geometrically true.
Therefore, the rate of change of the volume of the cube is approximately half of its surface area and the result is geometrically true.
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