The surface area S of a cube with edge length x is given by S(x) = 6x2 for x > 0

Question

The surface area S of a cube with edge length x is given by S(x) = 6x2 for x > 0. Suppose the cubes your company manufactures are supposed to have a surface area of exactly 42 square centimeters, but the machines you own are old and cannot always make a cube with the precise surface area desired. Write an inequality using absolute value that says the surface area of a given cube is no more than 3 square centimeters away (high or low) from the target of 42 square centimeters. Steps please?

Explanation / Answer

A cube has six faces; four faces along the vertical sides, one at the top, and another one at the bottom. That makes a total 6 faces. So there is the factor 6 in S ( x ) = 6 x ². Now, each face of a cube is a square and a square with all sides equal to x has a surface area of x • x = x ². A cube having six square faces has therefore the total surface area of S ( x ) = 6 x ². No more than 3 cm ² away from the target of S0 ( x ) = 42 cm ² means ……… a low of S1 ( x ) = S0 ( x ) - 3 cm ² = 42 cm ² - 3 cm ² = 39 cm ², and ……… a high of S2 ( x ) = S0 ( x ) + 3 cm ² = 42 cm ² + 3 cm ² = 45 cm ². That is ……… S1 ( x ) = S ( x ) = S2 ( x ) Using S1 ( x ) ……… S1 ( x ) = S ( x ) … --> … 39 cm ² = 6 x ² … --> … x ² = 39 cm ²/ 6 ……… x ² = 13 cm ²/ 2 … --> … x = v ( 13 cm ²/ 2 ) … --> … x = v ( 13 / 2 ) cm . We take the positive square root because x denotes the length od a side and there is no negative length. Using S2 ( x ) ……… S ( x ) = S2 ( x ) … --> … 6 x ² = 45 cm ² … --> … x ² = 45 cm ²/ 6 ……… x ² = 15 cm ²/ 2 … --> … x = v ( 15 cm ²/ 2 ) … --> … x = v ( 15 / 2 ) cm . The final answer is therefore ……… x lies within the interval [ v ( 13 / 2 ) , v ( 15 / 2 ) ] cm.

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