The question is
A rectangular piece of cardboard is 15 inches longer than it is wide. If 5-inch square corners are cut from each corner and the remaining piece folded up to form a box the volume of the box is 1250 cubic inches.
- Find the dimensions of the piece of cardboard
- Write a polynomial giving the surface area of the open box after the corners are cut away.
- using your dimensions from part a, what is the surface area of the open box.
First How i tried to solve this is by doing this
$$ V= L.W.H$$
$$ 1250 = (15-10)(w-10)(5) $$ Am i doing this right?
$$ 1250 = (15w^2 – 150w – 10w + 100) (5)$$
$$ 1250 = (15w^2 – 160w + 100) (5)$$
$$ 1250 = (75w^2 – 800W + 500) $$
$$ 1250 – 500 = 75w^2 – 800W$$
$$ 750 = 75w^2 – 800W$$
This is where i started having troubles, if i were to divide 800 and than divide whatever answer i got and squared root it than i would have gotten an number not correct to the answer. Can anyone help me to solve this? How do i find the dimensions of the piece of cardboard. For number 2 i don't know how to write the polynomial giving the surface area and as i was not able to do any of the 2 i can also not do number 3. Can anyone explain to me how to answer this problem? I have a test tomorrow and I have really bad grades right now.
Best Answer
It looks like you have a mistake in the beginning. It looks like you let W be the short side, but then the long side is 15+W because it says that the long side is 15 inches longer, so instead of 15-10 (or 15w-10 like in the image you wrote) you would write (15+W-10) and your polynomial would be correct. You should then get a quadratic so you should use the quadratic equation. So you would get
$1250 = (w+15-10)(w-10)(5)$
[dividing by 5]
$250 = (w+5)(w-10)$
$250 = w^2-10w+5w-50$
$250 = w^2 - 5w -50$
$w^2-5w-300 = 0$
then you can factor this as $(w-20)(w+15)$ so w = 20 or -15, but you can't have negative width so the answer is w = 20. You could also have done the quadratic equation at this point. Other than that mistake you were close!
As for the surface area, split up the box into the base of the box and the 4 sides. You know the base of the box has width $(w+5)(w-10)$ (see that from back up there the first line?). Then for the short side you know one side is 5 inches and the other is (w-10) so you get $5(w-10)$ for that side, and similarly you have $5(w+5)$ for the longer side. So your polynomial will be $(w+5)(w-10) + 2(5)(w-10) + 2(5)(w+5)$, notice that those sides appear twice in your box (they're opposite each other). Then you can plug in w = 20 into this polynomial to find the surface area of this box.