SOLUTION: Two young boys, Bob and Tom, are mowing grass one summer as a means of making money to save for their college education. They always split the work and proceeds equally. They have

Question 224750: Two young boys, Bob and Tom, are mowing grass one summer as a means of making money to save for their college education. They always split the work and proceeds equally. They have one lawnmower. They obtain a new job which involves mowing a 40' by 80' rectangular vacant lot. The owner wants it mowed in a collapsing pattern (begin mowing around the outside perimeter, with each pass moving closer to the center of the lot). Bob will mow first, turning the work over to Tom when he has mowed exactly half the area. Bob decides that he will stop at a uniform distance in from the perimeter on all sides, leaving a remaining rectangle for Tom to complete. At what uniform distance (in feet to 2 decimal places) should Bob stop mowing? Solve by creating and solving a quadratic equation.
I am lost! How do I solve this?

Answer by Earlsdon(6294) (Show Source): You can put this solution on YOUR website!
First, look at the entire rectangular vacant 40' by 80'lot.
Its area is:

sq.ft.
Half of this would be:

sq.ft.
A diagram would probably help you visualize the problem.
Draw a rectangle and, inside of that one, draw a smaller rectangle a uniform distance on all sides from the larger rectangle.
Now the larger rectangle L = 80' and W = 40" and its area is 3200 sq.ft.
The uniform inside distance between the two rectangles we'll call x, and this will be the number of feet at which Bob will stop mowing to turn the job over to Tom.
The area of the inside rectangle is to be 1600 sq.ft.
So we need to express the area of the inside rectangle in terms of the distance x.
The area of the interior rectangle can be expressed as:
and this is to be 1600 sq.ft., so...
Perform the indicated multiplication on the left side.
Subtract 1600 from both sides and rearrange.
Here's your quadratic equation. Factor out a 4 to ease the calculations a bit.
Now we need to solve the enclosed quadratic for x.
Since we can't factor this we'll use the quadratic formula to solve: where: a = 1, b = -60, and c = 400. Making the appropriate substitutions, we get:
Evaluate.

or
or Discard the extraneous (larger) solution.
Bob should stop mowing at a uniform distance of 7.64 feet.

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