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The volume of the interior of a rectangular storage bin is 240 cubic meters. If the perimeter of the interior base of the bin is 40 meters, which of the following could be the dimensions of the interior of the bin, in meters?

Indicate all such dimensions.

A. 6 by 8 by 5
B. 8 by 12 by 2.5
C. 9 by 11 by 2.4
D. 10 by 10 by 2.4

Volume has already been given. Volume lengthheightwidth = 240

The perimeter of a rectangle 2(height + width) = 40

Therefore, there are 3 unknown variables with two equations.

What should be the approach to the problem?

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    $\begingroup$ Have you tried anything at all? math.stackexchange.com/questions/ask $\endgroup$
    – Steve Kass
    Aug 12 at 15:16
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    $\begingroup$ 1. Calculate the volume. 2. If the volume is 240, calculate the perimeter of the bases $\endgroup$
    – Andrei
    Aug 12 at 15:36
  • $\begingroup$ There is going to be more than one answer, but only one of the possible answers will. $\endgroup$ Aug 12 at 18:08

1 Answer 1

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We know from the volume that $lwh=240$, and that $2(l+w)=40$ from the perimeter. We can rearrange the latter as $l=20-w$. Now, we substitute this expression for $l$ into the equation for volume. We get:$$(20-w)wh=240$$Dividing both sides by $h$, distributing, multiplying both sides by $-1$, and subtracting the right side of the equation from both sides, we get:$$w^2-20w+\frac{240}{h}=0$$Using the quadratic formula, we get:$$w=\frac{20\pm \sqrt{400-\frac{960}{h}}}{2}$$Plugging all of the possible answers into this relation, we find that only options B and D satisfy this, so they are the answers.

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    $\begingroup$ If you just love doing lots of extra work, then sure, this approach works. But just testing which of the four cases has the right volume and perimeter is a lot easier. $\endgroup$ Aug 13 at 22:53

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