The surface area S of a cube with edge length x is given by S(x) = 6x^2 for x >

Question

The surface area S of a cube with edge length x is given by S(x) = 6x^2 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. I just need to know how to write an equation for this. Thanks a bunch!

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 S ( x ) = 42 cm ² means
……… a low of S ( x ) = S ( x ) 3 cm ² = 42 cm ² 3 cm ² = 39 cm ², and
……… a high of S ( x ) = S ( x ) + 3 cm ² = 42 cm ² + 3 cm ² = 45 cm ².
That is ……… S ( x ) S ( x ) S ( x )
Using S ( x )
……… S ( x ) S ( x ) … --> … 39 cm ² 6 x ² … --> … x ² 39 cm ²/ 6
……… x ² 13 cm ²/ 2 … --> … x ( 13 cm ²/ 2 ) … --> … x ( 13 / 2 ) cm .
We take the positive square root because x denotes the length od a side and there
is no negative length.
Using S ( x )
……… S ( x ) S ( x ) … --> … 6 x ² 45 cm ² … --> … x ² 45 cm ²/ 6
……… x ² 15 cm ²/ 2 … --> … x ( 15 cm ²/ 2 ) … --> … x ( 15 / 2 ) cm .
The final answer is therefore
……… x lies within the interval [ ( 13 / 2 ) , ( 15 / 2 ) ] cm.

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