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Wed 22 Apr, 2009 05:08 pm
you have been hired by the school district as the designer of the new practice field. the school district recently acquired a 300 foot by 400 foot lot that they intend to use as a practice field. to accommodate all of the sports teams, the district decided yo have a walking path around the inside perimeter of the field you must determine the widest walking path that can be built so that there is still 100,000 square feet of practice field left. you must find the dimensions of the practice field, unknown walking path width, mathematical expressions for the unknown dimensions of the inner field, quadratic equation for the area of the practice field, definitions of variables used in the equation(s), work and solution for the quadratic equation, labeled graph of the quadratic equation, vertex of the parabola for this equation, and the dimensions of the walking path.
Have you drawn a diagram? That always helped me with these sorts of problems.
Draw a rectangle, then a smaller rectangle inside of it. The space between the two rectangles is the walking path, of width x.
The smaller rectangle with then have long sides of length 400-2x, and short sides of length 300-2x.
The area of the smaller rectangle is (400-2x)(300-2x), and needs to be at least 100,000 square feet. Solve for x.
Draw a rectangle with dimensions 400 by 300. Draw a second rectangle inside the first. This is the practice field and the area is 100000. The width of the path call x so dimensionally the inside rectangle has dimensions 400 minus twice the walk width by 300 minus twice the walk width. Set these dimension up using the walk width as the variable. Use the property of areas of rectangle and set it equal to 10000.
Rearrange to get a quadratic and solve for both solutions.d width. Use the sanity test to determine the solution and check your answer.
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