SOLUTION: I have to solve each problem then check my answer to be sure it is reasonable. Here's the problem: A ten-gallon aquarium holding African cichlids is 3 inches higher than it is w

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: I have to solve each problem then check my answer to be sure it is reasonable. Here's the problem: A ten-gallon aquarium holding African cichlids is 3 inches higher than it is w      Log On


   

Question 34925: I have to solve each problem then check my answer to be sure it is reasonable. Here's the problem:
A ten-gallon aquarium holding African cichlids is 3 inches higher than it is wide. It's length is 21 inches, and its volume is 2730 cubic inches. What are the height and width of the aquarium?
I have set it up to look like this:
(x-7)~2 + (x+1)~2 + x~2 then I have gotten that figured out to x~2-14x+49+x~2+2x+1=x~2 that is where I get stuck after subtracting the x~2 to get it on the same side as the rest of the equation.
Answer by mbarugel(146) About Me  (Show Source):
You can put this solution on YOUR website!
Hello!
Let's call X to the width of the aquarium. We know that its height must be X+3. Also, its length is 21. Volume is calculated as width*height*length. So we get the following equation:
X%28X%2B3%2921+=+2730
X%28X%2B3%29+=+2730%2F21+=+130
X%5E2+%2B+3X+=+130
X%5E2+%2B+3X+-+130+=+0
Now we can solve this quadratic equation using the standard procedure:
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B3x%2B-130+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%283%29%5E2-4%2A1%2A-130=529.

Discriminant d=529 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-3%2B-sqrt%28+529+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%283%29%2Bsqrt%28+529+%29%29%2F2%5C1+=+10
x%5B2%5D+=+%28-%283%29-sqrt%28+529+%29%29%2F2%5C1+=+-13

Quadratic expression 1x%5E2%2B3x%2B-130 can be factored:
1x%5E2%2B3x%2B-130+=+%28x-10%29%2A%28x--13%29
Again, the answer is: 10, -13. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B3%2Ax%2B-130+%29


As you can see the two solution are 10 and -13. We can disregard the -13 because width can't be negative. So we conclude that the width of the aquarium is 10 inches. Therefore, its height must be 13 inches.
You can check that 10%2A13%2A21+=+2730

I hope this helps!
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