SOLUTION: a farmer wants to set up a pigpen using 40 feet of fence to enclose a rectangular area of 51 square feet. what are the dimensions of the pigpen?

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Question 211995: a farmer wants to set up a pigpen using 40 feet of fence to enclose a rectangular area of 51 square feet. what are the dimensions of the pigpen?
Answer by drj(1380) About Me  (Show Source):
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A farmer wants to set up a pigpen using 40 feet of fence to enclose a rectangular area of 51 square feet. what are the dimensions of the pigpen?

1. Perimeter of a rectangle is 2x+2y=40 where x=one side of rectangle and y=the adjacent side of x. We can simplify this as x+y=20 where we divided 2 on both sides of the equation. That is,

%282x%2B2y%29%2F2=40%2F2=20

x%2By=20

y=20-x

2. Area=xy=51 where area of rectangle is height times width.

3. Substitute y in Step 1 into Step 2.

xy=x%2820-x%29=51 Multiplying the terms will yield: 20x-x%5E2=51

4. Put everything on the left side to the right. That is, add -20x+x^2 from both sides of the equation to make the left side equal to zero.

20x-x%5E2-20x%2Bx%5E2=+51%2Bx%5E2-20x

Simplifying will yield

0+=+51%2Bx%5E2-20x

Finally, the above equation is equivalent to

x%5E2-20x%2B51=0

4. Now this is just a Quadratic Equation so we can use the formula

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

where a=1, b=-20 and c=51

5. Use the following steps to solve the quadratic equation.

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-20x%2B51+=+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=%28-20%29%5E2-4%2A1%2A51=196.

Discriminant d=196 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--20%2B-sqrt%28+196+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-20%29%2Bsqrt%28+196+%29%29%2F2%5C1+=+17
x%5B2%5D+=+%28-%28-20%29-sqrt%28+196+%29%29%2F2%5C1+=+3

Quadratic expression 1x%5E2%2B-20x%2B51 can be factored:
1x%5E2%2B-20x%2B51+=+1%28x-17%29%2A%28x-3%29
Again, the answer is: 17, 3. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-20%2Ax%2B51+%29




6. The rectangular sides are 3 and 17. As a check 3*17=51 (Area is 51 square feet) and Perimeter is 2*(3+17)=40 ft. So everything checks out.

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