SOLUTION: A box with an open top is to be constructed from a rectangular piece of cardboard with the dimensions b=13 in by a= 20 in by cutting out equal squares of side x at each corner and
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Question 549404: A box with an open top is to be constructed from a rectangular piece of cardboard with the dimensions b=13 in by a= 20 in by cutting out equal squares of side x at each corner and then folding up the sides.
Express the volume of the box as a function of x.
Answer by oberobic(2304) (Show Source): You can put this solution on YOUR website!
When you cut the 'x' squares out of each corner, you are able to bold the sides up to form the box.
So, the depth of the box (or the height of the sides) = x.
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The 'a' sides will be reduced to 'a-2x' since you cut a square out of all 4 corners.
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Similarly, the 'b' sides will be reduced to 'b-2x' for the same reason.
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The volume of the box will be:
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x * (a-2x) * (b-2x) = volume
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Use FOIL to expand
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(x*a -2x^2)*(b-2x) = x*a*b - x*a*(-2x) -2x^2*b -2x^2*(-2x)
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x*a*b - x*a*(2x) -2x^2*b -2x^2*(-2x) = x*a*b -2*a*x^2 -2*b*x^2 + 4x^3
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Substitute a=20 and b=13.
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Notice 'x', 'a', and 'b' are measured in inches, so the result is in cubic inches (in^3).
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x*a*b -2*a*x^2 -2*b*x^2 + 4x^3 = 260x - 40x^2 -26x^2 + 4x^3
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260x -66x^2 + 4x^3 = 4x^3 -66x^2 +260x = volume
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Done.
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