Question

1. ryan left the science museum and drove south. gabriela left three hours later driving 42 km/h faster in an effort to catch up to him. after two hours gabriela finally caught up. find ryan’s average speed.
2. a submarine left hawaii two hours before an aircraft carrier. the vessels traveled in opposite directions. the aircraft carrier traveled at 25 mph for nine hours. after this time the vessels were 280 mi. apart. find the submarine’s speed.
3. chelsea left the white house and traveled toward the capital at an average speed of 34 km/h. jasmine left at the same time and traveled in the opposite direction with an average speed of 65 km/h. find the number of hours traveling needs to travel before they are 594 km apart.
4. jose left the airport and traveled toward the mountains. kayla left 2.1 hours later traveling 35 mph faster in an effort to catch up to him. after 1.2 hours kayla finally caught up. find jose’s average speed.

Explanation:

Problem 7:

Step1: Define Variables

Let Ryan's speed be \( x \) km/h. Then Gabriela's speed is \( x + 42 \) km/h.

Step2: Determine Time Traveled

Ryan traveled for \( 3 + 2 = 5 \) hours. Gabriela traveled for 2 hours.

Step3: Set Up Distance Equation

Since distance = speed × time and their distances are equal (Gabriela caught up), we have:
\( 5x = 2(x + 42) \)

Step4: Solve the Equation

Expand: \( 5x = 2x + 84 \)
Subtract \( 2x \): \( 3x = 84 \)
Divide: \( x = 28 \)

Step1: Define Variables

Let the submarine’s speed be \( x \) mph. The aircraft carrier’s speed is 25 mph.

Step2: Determine Time Traveled

The submarine traveled for \( 2 + 9 = 11 \) hours. The aircraft carrier traveled for 9 hours.

Step3: Set Up Distance Equation

Since they traveled in opposite directions, total distance = submarine’s distance + aircraft carrier’s distance:
\( 11x + 25 \times 9 = 280 \)

Step4: Solve the Equation

Calculate \( 25 \times 9 = 225 \). Then:
\( 11x + 225 = 280 \)
Subtract 225: \( 11x = 55 \)
Divide: \( x = 5 \)

Step1: Define Variables

Let the number of hours be \( t \). Chelsea’s speed is 34 km/h, so her distance is \( 34t \) km.
Isabelle’s speed is \( 34 + 45 = 79 \) km/h, and her time is \( t - \) (time difference, but since they start at same time? Wait, re-read: "Isabelle left at the same time and traveled in the opposite direction". Wait, distance between them is 594 km. So total distance = Chelsea’s distance + Isabelle’s distance:
\( 34t + 79t = 594 \)

Step2: Solve the Equation

Combine terms: \( 113t = 594 \)? Wait, no—wait, "average speed of 45 km/h" (wait, original text: "Isabelle left at the same time and traveled in the opposite direction with an average speed of 45 km/h". Wait, I misread earlier. Chelsea: 34 km/h, Isabelle: 45 km/h. So total distance: \( 34t + 45t = 594 \)
Combine: \( 79t = 594 \)? No, 34 + 45 = 79? Wait 34 + 45 = 79? 34 + 45 = 79. Then \( 79t = 594 \)? Wait 594 ÷ 79 ≈ 7.5? No, wait original text: "the number of hours traveling needs to travel before they are 594 km apart". So:
\( 34t + 45t = 594 \)
\( 79t = 594 \)? Wait 79×7=553, 79×8=632. No, maybe I misread speeds. Wait original: "Chelsea left the White House and traveled toward the capital at an average speed of 34 km/h. Isabelle left at the same time and traveled in the opposite direction with an average speed of 45 km/h. Find the number of hours traveling needs to travel before they are 594 km apart."
So correct equation: \( 34t + 45t = 594 \)
\( 79t = 594 \)? Wait 34 + 45 = 79. Then \( t = 594 / 79 ≈ 7.52 \)? No, 79×7=553, 594-553=41, so 7 + 41/79 ≈7.52. But maybe typo? Wait 34 + 45 = 79, 594 ÷79=7.518… Maybe I misread the speed. Wait original: "Isabelle left at the same time and traveled in the opposite direction with an average speed of 45 km/h". Wait 34 + 45 = 79. So:
\( t = 594 / (34 + 45) = 594 / 79 ≈ 7.52 \) hours. But maybe the speed is 45 km/h faster? Wait "with an average speed of 45 km/h" – no, "Isabelle left at the same time and traveled in the opposite direction with an average speed of 45 km/h" – wait Chelsea is 34, Isabelle is 45? Then total speed apart is 34 + 45 = 79. So time is 594 / 79 = 7.518 ≈7.52 hours. But maybe the problem has a typo, but following the math:

Step1: Define Variables

Let \( t \) = hours traveled.
Chelsea’s distance: \( 34t \)
Isabelle’s distance: \( 45t \) (opposite direction, so total distance is sum)
\( 34t + 45t = 594 \)

Step2: Solve

\( 79t = 594 \)
\( t = 594 / 79 ≈ 7.52 \) hours. But maybe the speed is 34 + 45? Wait no, the text says "Isabelle left at the same time and traveled in the opposite direction with an average speed of 45 km/h". So Isabelle’s speed is 45, Chelsea’s is 34. So total speed apart is 34 + 45 = 79. So time is 594 / 79 ≈7.52 hours.

Answer:

Ryan's average speed is 28 km/h.

Problem 8: