Quadratic Equations in Distance Problems Practice Problems
Set 1: Introduction to Quadratic Equations in Distance Problems  Traveling in One Direction or Stream
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Practice
 A flight from Dallas to Chicago is 800 miles. A plane flew with a 40mph tailwind from Dallas to Chicago. On the return trip, the plane flew against the same 40mph wind. The plane was in the air a total of 5.08 hours for the flight from Dallas to Chicago and the return flight. What would have been the plane’s speed without the wind?
 A flight from Houston to New Orleans faced a 50mph headwind, which became a 50mph tailwind on the return flight. The total time in the air was hours. The distance between Houston and New Orleans is 300 miles. How long was the plane in flight from Houston to New Orleans?
 A small motorboat traveled 15 miles downstream then turned around and traveled 15 miles back. The total trip took 2 hours. The stream’s speed is 4 mph. How fast would the boat have traveled in still water?
 A plane on a flight from Denver to Indianapolis flew with a 20mph tailwind. On the return flight, the plane flew into a 20mph headwind. The distance between Denver and Indianapolis is 1000 miles and the plane was in the air a total of hours. What would have been the plane’s average speed without the wind?
 A plane flew from Minneapolis to Atlanta, a distance of 900 miles, against a 30 mphheadwind. On the return flight, the 30mph wind became a tailwind. The plane was in the air for a total of hours. What would the plane’s average speed have been without the wind?
Solutions

Let r represent the plane’s average speed (in mph) with no wind. Then the average speed from Dallas to Chicago (with the tailwind) is r +40, and the average speed from Chicago to Dallas is r − 40 (against the headwind). The distance between Dallas and Chicago is 800 miles. The time in the air from Dallas to Chicago plus the time in the air from Chicago to Dallas is 5.08 hours. The time in the air from Dallas to Chicago is . The time in the air from Chicago to Dallas is . The equation to solve is . The LCD is ( r + 40)( r – 40).
The plane’s average speed without the wind would have been about 320 mph.

Let r represent the plane’s average speed without the wind. The average speed from Houston to New Orleans (against the headwind) is r – 50, and the average speed from New Orleans to Houston (with the tailwind) is r + 50. The distance between Houston and New Orleans is 300 miles. The time in the air from Houston to New Orleans is , and the time in the air from New Orleans to Houston is . The time in the air from Houston to New Orleans plus the time in the air from New Orleans to Houston is hours. The equation to solve is . The LCD is 4( r – 50)( r + 50).
The average speed of the plane without the wind was 350 mph. We want the time in the air from Houston to New Orleans: hour. The plane was in flight from Houston to New Orleans for one hour.

Let r represent the boat’s speed in still water. The average speed downstream is r + 4 and the average speed upstream is r – 4. The boat was in the water a total of 2 hours. The distance traveled in each direction is 15 miles. The time the boat traveled downstream is hours, and it traveled upstream hours. The time the boat traveled upstream plus the time it traveled downstream equals 2 hours. The equation to solve is . The LCD is ( r + 4)( r – 4).
The boat’s average speed in still water is 16 mph.

Let r represent the plane’s average speed without the wind. The plane’s average speed from Denver to Indianapolis is r + 20, and the plane’s average speed from Indianapolis to Denver is r – 20. The total time in flight is hours and the distance between Denver and Indianapolis is 1000 miles. The time in the air from Denver to Indianapolis is hours and the time in the air from Indianapolis to Denver is hours. The time in the air from Denver to Indianapolis plus the time in the air from Indianapolis to Denver is 5.5 hours. The equation to solve is . The LCD is (r + 20)( r – 20).
The plane would have averaged about 365 mph without the wind.

Let r represent the plane’s average speed without the wind. The plane’s average speed from Minneapolis to Atlanta (against the headwind) is r – 30. The plane’s average speed from Atlanta to Minneapolis (with the tailwind) is r + 30. The total time in the air is hours and the distance between Atlanta to Minneapolis is 900 miles. The time in the air from Minneapolis to Atlanta is hours, and the time in the air from Atlanta to Minneapolis is hours. The time in the air from Minneapolis to Atlanta plus the time in the air from Minneapolis to Atlanta is hours. The equation to solve is . The LCD is ( r – 30)( r + 30).
The plane’s average speed without the wind was 330 mph.
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