The surface area of a cube is increasing at a rate of [tex]$1 cm^2 s^{-1}$[/tex]. Find the rate of increase of the edge when the edge is 12 cm.
Answer (2)
Define variables: S (surface area), V (volume), x (edge length).
Express surface area and volume in terms of x : S = 6 x 2 , V = x 3 .
Differentiate both equations with respect to time t and use the given d t d S = 1 to find d t d x = 12 x 1 .
Substitute d t d x into the differentiated volume equation to find d t d V = 4 x , and evaluate at x = 12 to get 3 .
Explanation
Problem Analysis We are given that the surface area of a cube is increasing at a rate of 1 c m 2 / s . We want to find the rate of increase of the volume of the cube when the edge length is 12 cm.
Define Variables and Rates Let S be the surface area of the cube, V be the volume of the cube, and x be the edge length of the cube. We are given that d t d S = 1 c m 2 / s . We want to find d t d V when x = 12 c m .
Surface Area and its Rate of Change The surface area of a cube is given by S = 6 x 2 . Differentiating with respect to time t , we get d t d S = 12 x d t d x . Since d t d S = 1 , we have 1 = 12 x d t d x , so d t d x = 12 x 1 .
Volume and its Rate of Change The volume of a cube is given by V = x 3 . Differentiating with respect to time t , we get d t d V = 3 x 2 d t d x .
Relating Volume and Surface Area Rates Substituting d t d x = 12 x 1 into the equation for d t d V , we get d t d V = 3 x 2 ⋅ 12 x 1 = 4 x .
Calculate the Rate of Volume Increase When x = 12 , d t d V = 4 12 = 3 c m 3 / s .
Final Answer Therefore, the rate of increase of the volume when the edge is 12 cm is 3 c m 3 / s .
Examples
Imagine you're inflating a balloon that's perfectly cube-shaped. Knowing how fast the surface area is expanding helps you determine how quickly the balloon's volume is increasing. This is useful in manufacturing processes where controlling the rate of material deposition is crucial, such as in 3D printing or coating applications. By understanding these rates, you can ensure the final product meets the required specifications.
The rate of increase of the edge length of the cube when the surface area is increasing at a rate of 1 cm²/s and the edge length is 12 cm is 144 1 cm/s .
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