Question

1. across the united states, there are approximately 45,000 flights each day. in a certain simulation, the total number of flights that have begun across the united states n hours since midnight on a given day can be modeled using an arithmetic sequence, where 12:00 am is represented by n = 0. the total number of flights that had begun by 5 am that day was 11,750, and by 11 am there had been a total of 22,250 flights that had begun. based on the simulation, how many total flights had begun across the united states by 6 pm that day?

Explanation:

Step1: Calculate time interval from 11 AM to 6 PM

Find the number of hours from 11 AM ($n = 11$) to 6 PM ($n = 18$). The formula is $18 - 11 = 7$ hours.

Step2: Estimate flights per hour

Assume a uniform distribution of 45,000 daily flights. The number of flights per hour is $\frac{45000}{24}$ flights per hour.

Step3: Calculate flights from 11 AM to 6 PM

Multiply the number of hours (7) by the flights per hour $\frac{45000}{24}$. So the number of flights in this interval is $\frac{45000}{24}\times7$.

Step4: Calculate total flights by 6 PM

Add the number of flights that had begun by 11 AM (22250) to the number of flights from 11 AM to 6 PM. So $T=22250+\frac{45000}{24}\times7$ which equals 35375.

Answer:

We need to use the information about the number of flights at different - time points and the model to calculate the total number of flights. First, we know that at 5 AM ($n = 5$), the number of flights is part of the cumulative total. By 11 AM ($n = 11$), there had been 22,250 flights.

Let's assume the model for the number of flights $F(n)$ is a non - specified function of $n$ (number of hours since midnight). We know the cumulative number of flights up to 11 AM.

We want to find the number of flights from 11 AM to 6 PM ($n$ changes from 11 to 18).

We are not given the exact model for the number of flights per hour, but if we assume a constant rate of flight - start (which is not ideal but without more information...), we first note that the number of hours from 11 AM to 6 PM is $18 - 11=7$ hours.

We know that approximately 45,000 flights start each day. If we assume a uniform distribution of flights throughout the day (a big assumption), the number of flights per hour is $\frac{45000}{24}$ flights per hour.

The number of flights from 11 AM to 6 PM is approximately $\frac{45000}{24}\times7=\frac{315000}{24}=13125$

The total number of flights that had begun across the United States by 6 PM is the number of flights that had begun by 11 AM plus the number of flights from 11 AM to 6 PM.

So the total number of flights $T=22250 + 13125=35375$