Kirk and Montega have eccepted a job mowing the soccer field.They must mow a are

Question

Kirk and Montega have eccepted a job mowing the soccer field.They must mow a area 500 feet long and 400 feet wide they will bothmow half the area. Kirk will mow around the edge in a path of equalwidth until half the area is left. Write a quadratic equation thatcould be used to find the width x that Kirk should mow. The mowercan mow a path 5 feet wide, how many times should Kirk go aroundthe field? Kirk and Montega have eccepted a job mowing the soccer field.They must mow a area 500 feet long and 400 feet wide they will bothmow half the area. Kirk will mow around the edge in a path of equalwidth until half the area is left. Write a quadratic equation thatcould be used to find the width x that Kirk should mow. The mowercan mow a path 5 feet wide, how many times should Kirk go aroundthe field?

Explanation / Answer

The soccer field is 500 ft x 400 ft = 200,000 ft^2 in area. Eachwill mow half, or 100,000 ft^2. The inner rectangle has sides (500 - 2x) ft and (400 - 2x) ft. Theinner rectangle is therefore (500 - 2x)(400 - 2x) = 200,000 - 1800x+ 4x^2 ft^2 in area. Montega is mowing the inner rectangle, whichis half of the total, or 100,000 ft^2. (From here on out, I amdropping the "ft^2" for simplicity, but remember that the area isin ft^2 and the length and width are in ft.) Therefore 200,000 -1800x + 4x^2 = 100,000. Subtract 100,000 from both sides: 100,000 -1800x + 4x^2 = 0. Rearrange: 4x^2 - 1800x + 100,000 = 0. Divideboth sides by 4: x^2 - 450x + 25,000 = 0. Solve for x using complete-the-square or the quadratic formula: x^2 - 450x + 25000 = 0 x^2 - 450x = -25000 x^2 - 450x + 50625 = 50625 - 25000 = 25625 (x-225)^2 = (approx +/-160)^2 x - 225 = approx +/- 160 x = approx 65 ft or approx 385 ft. Which of these makes sense? Does x = 385 ft make sense? No, becausethe inner rectangle has sides of (500 - 2x) ft and (400 - 2x) ft,and x = 385 ft would make these negative. So the answer must be x =65 ft (approx.). Plug in the answer to check: Inner rectangle has area (500 - 2x)(400 - 2x) = 370(270), which is99900, which is close to 100,000.

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