An Air Force jet is going on a long-distance flight and will need to be refueled in midair over the Pacific Ocean. A refueling plane that carries fuel can travel much farther than the jet but flies at a slower speed. The refueling plane and jet will leave from the same base, but the refueling plane will leave 3 hours before the jet. The jet will fly at 800 mph and the refueling plane will fly at 530 miles per hour. a) How long after the jet takes off will the two planes meet? b) How far from the base will the refueling take place? a) How long after the jet takes off will the two planes meet? hours (Type an integer or decimal rounded to four decimal places as needed.)

To find out how long after the jet takes off the two planes will meet, we can step through the problem methodically. The refueling plane leaves 3 hours before the jet. In that time, it would have traveled a distance of: Distance = Speed × Time = 530 mph × 3 hours = 1590 miles. Once the jet takes off, both planes will be moving toward each other. The relative speed at which they approach one another is: Relative Speed = Speed of Jet + Speed of Refueling Plane = 800 mph + 530 mph = 1330 mph. Now, we need to find out how long it will take for the jet to catch up to the refueling plane. They will meet when the distance the jet travels (D) equals the distance the refueling plane had initially covered plus the distance it continues to cover while being pursued. Let \( t \) be the time (in hours) after the jet takes off until they meet. The jet will cover \( 800t \) miles in that time, while the refueling plane, which started 1590 miles ahead, will cover \( 530t \) miles additionally. Therefore, we can set up the equation: Distance that the jet travels = Distance covered by the refueling plane initially + Distance the refueling plane covers during the jet's flight: \[ 800t = 1590 + 530t \] Now, we will solve for \( t \): \[ 800t - 530t = 1590 \] \[ 270t = 1590 \] \[ t = \frac{1590}{270} \approx 5.8889 \text{ hours} \] So, after the jet takes off, it will take approximately **5.8889 hours** for the two planes to meet. Now for part b: To find how far from the base the refueling will take place, we can use the time it took for the jet to meet the refueling plane. Using the jet's speed: Distance = Speed × Time: \[ \text{Distance from the base} = 800 \text{ mph} × 5.8889 \text{ hours} \approx 4711.1 \text{ miles} \] So, the refueling will take place **approximately 4711.1 miles** from the base.