SOLUTION: The length of Jim's leader is always 2 feet shorter than the length of his fishing rod. If the product of the length of the leader and the length of the rod equals the strength of

Algebra ->  College  -> Linear Algebra -> SOLUTION: The length of Jim's leader is always 2 feet shorter than the length of his fishing rod. If the product of the length of the leader and the length of the rod equals the strength of      Log On


   

Question 238098: The length of Jim's leader is always 2 feet shorter than the length of his fishing rod. If the product of the length of the leader and the length of the rod equals the strength of his line in pounds and he uses a 16-lb line, what is the length of his leader?
Answer by Stitch(470) About Me  (Show Source):
You can put this solution on YOUR website!
Let X = the length of the rod
Let Y = the length of the leader
Equation 1: Y+=+X+-+2
Equation 2: X+%2A+Y+=+16
Using equation 1, plug (X - 2) into equation 2 for Y
X+%2A+Y+=+16
X+%2A+%28X+-+2%29+=+16 Rewrite the equation
X%5E2+-+2X+=+16 Subtract 16 form both sides
X%5E2+-+2X+-+16+=+0 Then you need to use 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-2X%2B-16+=+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=%28-2%29%5E2-4%2A1%2A-16=68.

Discriminant d=68 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--2%2B-sqrt%28+68+%29%29%2F2%5Ca.

X%5B1%5D+=+%28-%28-2%29%2Bsqrt%28+68+%29%29%2F2%5C1+=+5.12310562561766
X%5B2%5D+=+%28-%28-2%29-sqrt%28+68+%29%29%2F2%5C1+=+-3.12310562561766

Quadratic expression 1X%5E2%2B-2X%2B-16 can be factored:
1X%5E2%2B-2X%2B-16+=+1%28X-5.12310562561766%29%2A%28X--3.12310562561766%29
Again, the answer is: 5.12310562561766, -3.12310562561766. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-2%2Ax%2B-16+%29

Since you can not have a negative length, the 5.123 is the only rational answer.
But remember that 5.123 is the length of the rod.
To find the length of the leader, plug 5.123 into equation 1 for X
Equation 1: Y+=+X+-+2
Y+=+5.123+-+2
highlight+%28Y+=+3.123%29
The leader = 3.123ft