An airplane flew 3060
 miles. It later took off and flew another 4500
 miles at a average speed of 60
 miles per hour slower than the first trip. The second trip took 4
 hours more time than the first. Find the number of minutes for the first trip.

To solve this problem, let's break it down step by step.

Let:
- \( r_1 \) = the average speed (in miles per hour) for the first trip,
- \( r_2 \) = the average speed for the second trip,
- \( t_1 \) = the time (in hours) taken for the first trip,
- \( t_2 \) = the time taken for the second trip.

### Step 1: Time equation for each trip

- For the first trip: 
  \[
  \text{Distance} = \text{Speed} \times \text{Time} \implies t_1 = \frac{3060}{r_1}
  \]

- For the second trip:
  \[
  t_2 = \frac{4500}{r_2}
  \]

### Step 2: Relationship between the speeds

From the problem, we know that the speed for the second trip is 60 miles per hour slower than the speed for the first trip:
\[
r_2 = r_1 - 60
\]

### Step 3: Relationship between the times

We are also told that the second trip took 4 hours longer than the first trip:
\[
t_2 = t_1 + 4
\]

### Step 4: Set up equations

Substitute the expressions for \( t_1 \) and \( t_2 \) into the time relationship equation:

\[
\frac{4500}{r_2} = \frac{3060}{r_1} + 4
\]

Now, replace \( r_2 \) with \( r_1 - 60 \):
\[
\frac{4500}{r_1 - 60} = \frac{3060}{r_1} + 4
\]

### Step 5: Solve the equation

Now we solve this equation to find \( r_1 \), the speed of the first trip.

Multiply both sides of the equation by \( r_1(r_1 - 60) \) to eliminate the denominators:

\[
4500r_1 = 3060(r_1 - 60) + 4r_1(r_1 - 60)
\]

Expanding both sides:

\[
4500r_1 = 3060r_1 - 3060 \times 60 + 4r_1^2 - 240r_1
\]

Simplify the equation:

\[
4500r_1 = 3060r_1 - 183600 + 4r_1^2 - 240r_1
\]

Combine like terms:

\[
0 = 4r_1^2 - 240r_1 + 3060r_1 - 4500r_1 - 183600
\]
\[
0 = 4r_1^2 - 240r_1 + 3060r_1 - 4500r_1 - 183600
\]
\[


An airplane flew 3060 miles. It later took off and flew another 4500 miles at an average speed of 60 miles per hour slower than the first trip. The second trip took 4 hours more time than the first. Find the number of minutes for the first trip.

Learn how to solve a real-world airplane travel problem involving time, speed, and distance. This detailed solution covers linear and quadratic equations, breaking down how to find the time for a 3060-mile trip. Suitable for students in Grades 9-11.

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