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 cube when the edge is 12 cm.
Answer (1)
Establish the surface area formula for a cube: S = 6 x 2 .
Differentiate with respect to time t to relate the rates: 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.
Solve for d t d x to find the rate of increase of the edge length: 72 5 cm/s.
Explanation
Problem Setup We are given that the surface area of a cube is increasing at a rate of 10 c m 2 / s , and 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 of the cube.
Surface Area Formula The surface area of a cube is given by the formula: S = 6 x 2
Differentiating with Respect to Time Now, we differentiate both sides of the equation with respect to time t to relate the rates of change of the surface area and the edge length: d t d S = d t d ( 6 x 2 ) Using the chain rule, we get: d t d S = 12 x d t d x
Substituting Given Values We are given that d t d S = 10 c m 2 / s and x = 12 c m . Substitute these values into the equation: 10 = 12 ( 12 ) d t d x
Solving for Rate of Increase of Edge Length Now, solve for d t d x :
d t d x = 12 ( 12 ) 10 = 144 10 = 72 5
Final Answer Therefore, the rate of increase of the edge length when the edge is 12 cm is 72 5 cm/s.
Examples
Imagine you're inflating a balloon that's perfectly cube-shaped. Knowing how fast the surface area is increasing helps you determine how quickly the sides are growing. This is useful in manufacturing processes where you need to control the growth rate of materials to achieve specific dimensions or properties. Understanding these rates ensures precision and quality in production.
