Question 531784
The volume of a rectangular box is 6 ft^3 more than twice the volume of a cube. The length of the box is 2 ft more than the length of an edge of the cube its width is equal to the length of an edge of the cube; and its height is 1 ft more than the length of an edge of the cube. What are the dimensions of the box and the cube?
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let x = the side of the cube
Write an equation for each statement:
"The volume of a rectangular box is 6 ft^3 more than twice the volume of a cube."
L * W * H = 2x^3 + 6
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" The length of the box is 2 ft more than the length of an edge of the cube"
L = x + 2
:
"its width is equal to the length of an edge of the cube;"
W = x
:
"its height is 1 ft more than the length of an edge of the cube."
H = x + 1
:
Substituting for L,W,H
(x+2)*x*(x+1) = 2x^3 + 6
x(x^2 + x + 2x + 2) = 2x^3 + 6 
x(x^2 + 3x + 2) = 2x^3 + 6
x^3 + 3x^2 + 2x = 2x^3 + 6
Combine like terms on the left
x^3 - 2x^3 + 3x^2 + 2x - 6 = 0
-x^3 + 3x^2 + 2x - 6 = 0
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Graph this equation, there are two positive solutions but x=3 is an integer
{{{ graph( 300, 200, -6, 5, -10, 10, -x^3 + 3x^2 + 2x - 6 ) }}}
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What are the dimensions of the box and the cube?
 x = 3 inches the side of the cube
then
 L = 3+2 = 5 inches
 W = 3 inches
 H = 3+1 = 4 inches
:
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Check this
5*3*4 = 2(3^3) + 6
60 = 2(27) + 6