An open box is to be made out of a 6 inch by 16 inch piece...

An open box is to be made out of a 6-inch by 16-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box that has the largest volume. Dimensions of the bottom of the box: ______ x ________ Height: __________

Transcript

00:01 Alright, so we're gonna construct an open box out of a 16 by 6 inch piece of cardboard and we're gonna cut some squares out of the corners for equal squares, we're gonna bend up the sides.

00:16 So down here i've got the i've got the piece of cardboard and you see we'll just cut out those corners and we'll fold up the sides and this is what we'll end up with and we want to find the dimensions of the resulting box it has the largest volume.

00:33 Okay, and then we're gonna put that there and hopefully hopefully we'll get that done.

00:40 So i drew this i drew this figure to represent cutting the squares out of the box and so each one of these little lengths is x so now imagine if you will this whole thing is 6 right because it's 6 by 16 but we're gonna cut out two of the squares so this bit right here is gonna end up being the width and you can see it's down here the width in my other picture because once i fold it up this length will be the same as this length so here's my width 6 minus 2 x now one thing you notice 6 minus 2 x i don't want that to be 0 so my squares can't be bigger than 3 inches because if they were 3 inches we'd have a 3 inches here and 3 inches here 3 plus 3 is 6 6 minus 6 is 0 we wouldn't have we would just we had cut out we'd have a little strip right here but none of this would it all be gone.

01:43 So that's no good so we have to that tells us the values for the for x are going to be between 0 and 3 so the same reasoning goes with 16 here i've got the whole thing is 16 and then i'm taking out two of the x's and so there's my over here on my diagram to the right.

02:11 There's my 16 minus 2 x so if you look at my diagram here with the gray box i've got 6 minus 2 x times 16 minus 2 x times x x is the height because again we fold it up this square and this square is x.

02:27 So our length of x so there's my volume and i multiplied it out and we've got a cubic now i thought i'm gonna graph it but i'm only gonna graph it for x values between 0 and 3 because over here we just said that we can't have anything bigger than 3 and we don't obviously don't want to have a square of 0 then we haven't cut anything out and we don't have a box or we don't yeah, we don't have a box.

02:57 We just still have our cardboard sheet...

Question

An open box is to be made out of a 6-inch by 16-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box that has the largest volume. Dimensions of the bottom of the box: x Height: ____

Question

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