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An open box is to be made from a rectangular piece of material, three times as long as it is wide, by cutting two inch squares from each corner and turning up the sides. The volume of the finished box is to be 312 cubic inches. Find the size of the original piece of material. V = lwh

Answer 1

Hi Kanyen,

If I'm understanding you correctly: the original rectangle has length (l) and width (w). But in order to make a 3-D rectangular box, we have to cut squares out of each of the original rectangle's corners. Those squares are 2x2 inches (in.), and thus this will decrease the rectangle's width (w) by 2 in. on one side, and 2 in. on the other. Therefore the (w) decreases by 2+2=4 in total. The same idea relates to the length (l) of the rectangle. The total (l) that will make the box will be decreased also by 2=2=4.

So, since we know the area of original rectangle (A_R) = length (l) x width (w) = lw

Also we're told a very helpful bit of information that the rectangle's length (l) is 3 times as long as its width (w), so l = 3w

Now since our final 3-D rectangular box is exactly 312 in.³, and Volume of the rectangular box (V_B)=lwh. The height (h) is just 2 in. because that's the end flap that we cut into our original rectangle.

Now we must make new variables for our box dimensions: V_B = 312 in.³ , but the original l and w have decreased each by 4in. to make the flaps for the 3-D box. So let's use L and W as the new shortened dimensions. L = l - 4, and W = w - 4, and our h remains 2 in. Now remember above that our original l = 3w... so now let's plug in "3w" for l --> L = 3w - 4

Now to setting up our Volume equation to solve: V_B = 312 = LxWxh = LWh, and we just plug in what we know...

312 = (3w-4)x(w-4)x2 --> divide both sides by 2, then multiply out, so: 312/2 = [(3w² - 12w - 4w + 16)x2]/2

--> 156 = 3w² - 12w - 4w + 16 = 3w² - 16w + 16, so 156 = 3w² - 16w + 16, now we have a quadratic function that in order to be solved, needs to be set equal to 0 and factored out to solve for w: 3w² - 16w + 16 -156 = 156-156

--> 3w² - 16w -140 = 0 --> factor out the 1st and last terms to find the best fit to get us that middle coefficient of 16w (16) from factors of the 1st coefficient (3) and the last coefficient (-140), so 3: 1x3; 140: 1x140 (highly unlikely), 2x70 (still 70 is too high to get us to 16 for the middle term), 4x35, 7x20, and 10x14. If you play around with those you'll see that (3 x -10) + (1 x 14) gives us what we need, because -30 + 14 = -16 --> therefore our factors should look like: (3w + 14)(w - 10) because F.O.I.L. that and we get our 3w² - 16w - 140.

Now back to our solving... so (3w + 14)(w - 10) = 0, let's solve for each: 3w + 14 = 0 --> 3w = -14 --> w = -14/3 =-4.67; w - 10 = 0 --> w = 10\

*the only possible solution for (w) now must be 10, because we cannot have a negative width measurement, as is -4.67

Next we plug in to our original rectangular area AR = lxw, and l = 3w, so now that w = 10, so l = 3(10) = 30, and

A_R = 10x30 = 300 in.²

Hope that helped you to understand Kalyen, without confusing you more!

Best,

Clyde W.

https://www.wyzant.com/Tutors/Biology_Prof