SOLUTION: An open box is to be made from a 10 cm by 20 cm rectangular piece of cardboard by cutting a square from each corner and folding up the sides. The area of the bottom of the box is

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Question 118426: An open box is to be made from a 10 cm by 20 cm rectangular piece of cardboard by cutting a square from each corner and folding up the sides. The area of the bottom of the box is 96 cm squared. What is the length of the sides of the squares that are cut from the corners?
I really need help solving this algebraically. If you could help that would be amazing!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
If you draw a picture, you can see that the dimensions of the bottom of the box are 20-2x by 10-2x


Photobucket


So the area of the bottom is A=%2820-2x%29%2810-2x%29


96=%2820-2x%29%2810-2x%29 Now plug in the given area A=96


96=200-60x%2B4x%5E2 Foil


0=200-60x%2B4x%5E2-96 Subtract 96 from both sides


0=104-60x%2B4x%5E2 Combine like terms


0=4x%5E2-60x%2B104 Rearrange the terms


Let's use the quadratic formula to solve for x:


Starting with the general quadratic

ax%5E2%2Bbx%2Bc=0

the general solution using the quadratic equation is:

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29



So lets solve 4%2Ax%5E2-60%2Ax%2B104=0 ( notice a=4, b=-60, and c=104)




x+=+%28--60+%2B-+sqrt%28+%28-60%29%5E2-4%2A4%2A104+%29%29%2F%282%2A4%29 Plug in a=4, b=-60, and c=104



x+=+%2860+%2B-+sqrt%28+%28-60%29%5E2-4%2A4%2A104+%29%29%2F%282%2A4%29 Negate -60 to get 60



x+=+%2860+%2B-+sqrt%28+3600-4%2A4%2A104+%29%29%2F%282%2A4%29 Square -60 to get 3600 (note: remember when you square -60, you must square the negative as well. This is because %28-60%29%5E2=-60%2A-60=3600.)



x+=+%2860+%2B-+sqrt%28+3600%2B-1664+%29%29%2F%282%2A4%29 Multiply -4%2A104%2A4 to get -1664



x+=+%2860+%2B-+sqrt%28+1936+%29%29%2F%282%2A4%29 Combine like terms in the radicand (everything under the square root)



x+=+%2860+%2B-+44%29%2F%282%2A4%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%2860+%2B-+44%29%2F8 Multiply 2 and 4 to get 8

So now the expression breaks down into two parts

x+=+%2860+%2B+44%29%2F8 or x+=+%2860+-+44%29%2F8

Lets look at the first part:

x=%2860+%2B+44%29%2F8

x=104%2F8 Add the terms in the numerator
x=13 Divide

So one answer is
x=13



Now lets look at the second part:

x=%2860+-+44%29%2F8

x=16%2F8 Subtract the terms in the numerator
x=2 Divide

So another answer is
x=2

So our possible solutions are:
x=13 or x=2

However, since 13 is greater than 10, the solution x=13 is not possible since the square cutout cannot have a longer length than one of the sides.


So our only solution is x=2



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Check:


Remember, the area is

A=%2820-2x%29%2810-2x%29


96=%2820-2%282%29%29%2810-2%282%29%29 Plug in the given area A=96 and x=2


96=%2820-4%29%2810-4%29 Multiply


96=%2816%29%286%29 Subtract


96=96 Multiply. Since the two sides of the equation are equal, this verifies our answer.