The surface area of a cube is increasing at a rate of [tex]$10 cm^2 s^{-1}$[/tex]. Find the rate of increase of the edge when the edge is 12 cm.
Answer (1)
Write the formula for the surface area of a cube: S = 6 x 2 .
Differentiate both sides with respect to time t : d t d S = 12 x d t d x .
Substitute the given values d t d S = 10 and x = 12 into the equation: 10 = 12 ( 12 ) d t d x .
Solve for d t d x : d t d x = 144 10 = 72 5 .
72 5
Explanation
Problem Analysis We are given that the surface area of a cube is increasing at a rate of 10 c m 2 / s . We want to find the rate of increase of the edge length when the edge length is 12 cm. Let S be the surface area of the cube and x be the edge length.
Surface Area Formula The surface area of a cube is given by the formula S = 6 x 2 . We need to find d t d x when x = 12 cm and d t d S = 10 c m 2 / s .
Differentiation Differentiate both sides of the equation S = 6 x 2 with respect to time t using the chain rule: d t d S = d t d ( 6 x 2 ) = 12 x d t d x
Substitution Now, substitute the given values into the equation: d t d S = 10 and x = 12 :
10 = 12 ( 12 ) d t d x 10 = 144 d t d x
Solving for dx/dt Solve for d t d x :
d t d x = 144 10 = 72 5 The rate of increase of the edge length is 72 5 cm/s.
Final Answer Therefore, the rate of increase of the edge length of the cube when the edge is 12 cm is 72 5 cm/s.
Examples
Imagine you're designing expanding metallic cubes for art installations. Knowing how the edge length increases with surface area is crucial. This calculation helps determine how quickly the cube expands, ensuring it meets aesthetic and structural requirements during inflation. Understanding these rates allows precise control over the installation's visual impact and safety.
