SOLUTION: A 30 inch by 40 inch countertop for a work island is to be covered with green ceramic tiles, except for a border of uniform width, if the area covered by green tiles is 704 square

Algebra ->  Exponents -> SOLUTION: A 30 inch by 40 inch countertop for a work island is to be covered with green ceramic tiles, except for a border of uniform width, if the area covered by green tiles is 704 square       Log On


   

Question 149810: A 30 inch by 40 inch countertop for a work island is to be covered with green ceramic tiles, except for a border of uniform width, if the area covered by green tiles is 704 square inches (in^2 ) , then how wide is the border?
Found 2 solutions by jim_thompson5910, josmiceli:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let x=width of border.

First, let's draw the picture.

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note: the lengths 40-2x and 30-2x are due to the subtraction of 2 "x" lengths. Take note that there are two dashed lines per side that are not part of the green tiles.


From the picture, we see that the actual counter top is 40-2x by 30-2x. So the length is L=40-2x and the width is W=30-2x


Now remember, the area of a rectangle is

A=L%2AW

704=%2840-2x%29%2830-2x%29 Plug in the given area of the green tiles A=704, L=40-2x and W=30-2x


704=1200-140x%2B4x%5E2 FOIL


0=1200-140x%2B4x%5E2-704 Subtract 704 from both sides.


0=4x%5E2-140x%2B496 Combine like terms.


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%28-140%29+%2B-+sqrt%28+%28-140%29%5E2-4%284%29%28496%29+%29%29%2F%282%284%29%29 Plug in a=4, b=-140, and c=496


x+=+%28140+%2B-+sqrt%28+%28-140%29%5E2-4%284%29%28496%29+%29%29%2F%282%284%29%29 Negate -140 to get 140.


x+=+%28140+%2B-+sqrt%28+19600-4%284%29%28496%29+%29%29%2F%282%284%29%29 Square -140 to get 19600.


x+=+%28140+%2B-+sqrt%28+19600-7936+%29%29%2F%282%284%29%29 Multiply 4%284%29%28496%29 to get 7936


x+=+%28140+%2B-+sqrt%28+11664+%29%29%2F%282%284%29%29 Subtract 7936 from 19600 to get 11664


x+=+%28140+%2B-+sqrt%28+11664+%29%29%2F%288%29 Multiply 2 and 4 to get 8.


x+=+%28140+%2B-+108%29%2F%288%29 Take the square root of 11664 to get 108.


x+=+%28140+%2B+108%29%2F%288%29 or x+=+%28140+-+108%29%2F%288%29 Break up the expression.


x+=+%28248%29%2F%288%29 or x+=++%2832%29%2F%288%29 Combine like terms.


x+=+31 or x+=+4 Simplify.


So the possible answers are x+=+31 or x+=+4


However, we need to see if they generate reasonable lengths and widths


Let's check the solution x+=+31


L=40-2x Go back to the length equation


L=40-2%2831%29 Plug in x+=+31


L=40-62 Multiply


L=-22 Subtract


Since a negative length is not possible, this means that the value x+=+31 is a reasonable solution.


----------------------------



Let's check the solution x+=+4


L=40-2x Go back to the length equation


L=40-2%284%29 Plug in x+=+4


L=40-8 Multiply


L=32 Subtract

So we get a reasonable length with x+=+4

--------

W=30-2x Go back to the width equation


W=30-2%284%29 Plug in x+=+4


W=30-8 Multiply


W=22 Subtract

So we also get a reasonable width with x+=+4


===============================================

Answer:

So the only solution is x+=+4


So the width of the border is 4 inches

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let the width of the border = x
%2840+-+x%29%2830+-+x%29+=+704
1200+-+70x+%2B+x%5E2+=+704
x%5E2+-+70x+%2B+496+=+0
Use quadratic formula to solve
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
a+=+1
b+=+-70
c+=+496
x+=+%28-%28-70%29+%2B-+sqrt%28+%28-70%29%5E2-4%2A1%2A496+%29%29%2F%282%2A1%29+
x+=+%2870+%2B-+sqrt%284900+-+1984+%29%29+%2F+2+
x+=+%2870+%2B-+sqrt%28+2916+%29%29+%2F+2+
x+=+%2870+%2B-+54%29+%2F+2+
x+=+62 this is too large to use
x+=+8 in answer
check answer:
%2840+-+x%29%2830+-+x%29+=+704
%2840+-+8%29%2830+-+8%29+=+704
32%2A22+=+704
704+=+704
OK