In a race, a long distance runner took 3 minutes and 45 seconds to run the first kilometer, and in each succeeding kilometer, 12 seconds more than the preceding kilometer. Find the total running distance after 40 minutes.

first one = a = 3*60 = 180 seconds

change each kilometer = d = 12 seconds
so time for each kilometer = 180 + 12 ( n-1)
arithmetic sequence
sum of times for n kilometers = (n/2) [ 2a + (n-1)d ]
( google arithmetic sequence math is fun)
so
40*60 seconds = (n/2) [ 360 + (n-1)(12) ]
2400 = 180 n + 6 n^2 - 6 n
6 n^2 + 174 n -2400 = 0
n^2 + 29 n - 400 = 0
n = 10.2 or -38.2 use 10.2 kilometers

To find the total running distance after 40 minutes, we need to first determine how many kilometers the runner covers in 40 minutes, given the information provided.

Let's break it down:

First, we need to convert the runner's initial time of 3 minutes and 45 seconds to seconds. There are 60 seconds in a minute, so:

3 minutes * 60 seconds/minute = 180 seconds
180 seconds + 45 seconds = 225 seconds

Therefore, the runner took 225 seconds to run the first kilometer.

Next, we need to find the time it takes the runner to complete each subsequent kilometer. We know that in each succeeding kilometer, the runner takes 12 seconds more than the previous kilometer.

To save time, we can use arithmetic progression because the time taken forms an arithmetic progression:

The first term, a₁ = 225 seconds
The common difference, d = 12 seconds

Now, let's find an equation for the nth term of the arithmetic progression, where n represents the kilometer:

aₙ = a₁ + (n-1)d

We can use this equation to find the time taken for each kilometer:

a₂ = 225 + (2-1) * 12 = 225 + 12 = 237 seconds
a₃ = 225 + (3-1) * 12 = 225 + 24 = 249 seconds

We can continue this pattern for each kilometer, but we want to find the number of kilometers the runner covers in 40 minutes. We know that:

40 minutes = 40 * 60 = 2400 seconds

Now, let's find the number of terms (kilometers) the runner completes in 2400 seconds using the formula for the nth term:

aₙ = a₁ + (n-1)d

2400 = 225 + (n-1) * 12
2400 - 225 = (n-1) * 12
2175 = (n-1) * 12

To simplify the equation, let's divide both sides by 12:

2175/12 = n - 1
181.25 = n - 1

Adding 1 on both sides, we get:

181.25 + 1 = n
n ≈ 182.25

Since we can't have a fraction of a kilometer, we will round down to the nearest whole number.

Therefore, in 40 minutes, the runner covers approximately 182 kilometers.

So the total running distance after 40 minutes is 182 kilometers.