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A rectangular box has the dimensions shown in the diagram. The volume of the box is given by the function V(x) = x3 – 4x, where x is the height in inches.

What is the height of the box if the volume is 15 in.3?


There is a rectangular box with length x+2 in. , width x-2 in. , and height x in.


height = in.

A rectangular box has the dimensions shown in the diagram The volume of the box is given by the function Vx x3 4x where x is the height in inchesWhat is the hei class=

Respuesta :

MrRoyal

Answer:

[tex]Height =3[/tex]

Step-by-step explanation:

Given

[tex]Length = x + 2[/tex]

[tex]Width = x - 2[/tex]

[tex]Height = x[/tex]

[tex]Volume = 15[/tex]

Required

Find the height

The volume is calculated as thus:

[tex]Volume = Length * Width * Height[/tex]

This gives

[tex]15 = (x + 2) * (x - 2) * x[/tex]

Apply difference of two squares

[tex]15 =(x^2 - 4 ) * x[/tex]

[tex]15 = x^3 - 4x[/tex]

Subtract 15 from both sides

[tex]x^3 - 4x - 15 = 0[/tex]

Using a calculator to factorize, we have:

[tex](x-3)(x^2+3x+5) = 0[/tex]

Split

[tex]x - 3 = 0[/tex] or [tex]x^2 + 3x + 5 = 0[/tex]

[tex]x = 3[/tex] or  [tex]x^2 + 3x + 5 = 0[/tex]

x has complex roots for [tex]x^2 + 3x + 5 = 0[/tex]

Hence:

[tex]x = 3[/tex]

Recall that:

[tex]Height = x[/tex]

[tex]Height =3[/tex]

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