Question

We have given the timings of the gold medalists in the 400-metre race from the time the event was included in the Olympics, in the table below. Construct a mathematical model relating the years and timings. Use it to estimate the timing in the next Olympics.

| Year | Timing (in seconds) |
| 1964 | 52.01 |
| 1968 | 52.03 |
| 1972 | 51.08 |
| 1976 | 49.28 |
| 1978 | 48.88 |
| 1984 | 48.83 |
| 1988 | 48.65 |
| 1992 | 48.83 |
| 1996 | 48.25 |
| 2000 | 49.11 |
| 2004 | 49.41 |

Answer 1

Formulation: In the figure, the times of the gold medalist of 400 metres race are given of the Olympics (1964 - 2004). We take 1964 as zeroth years and write 1 for 1968, 2 for 1972 and 3 for 1976. We prepare a new table.

| Year | Timing (in seconds) |
| 0 | 52.01 |
| 1 | 52.03 |
| 2 | 51.08 |
| 3 | 49.28 |
| 4 | 48.88 |
| 5 | 48.83 |
| 6 | 48.65 |
| 7 | 48.83 |
| 8 | 48.25 |
| 9 | 49.11 |
| 10 | 49.41 |

The reduction in timings of gold medalist in 400 metres rave in Olympics given in the following table.

| Year | Timings | Change in timings |
| 0 | 52.01 | 0 |
| 1 | 52.03 | +0.02 |
| 2 | 51.08 | -0.95 |
| 3 | 49.28 | +1.80 |
| 4 | 48.88 | -0.40 |
| 5 | 48.3 | -0.05 |
| 6 | 48.65 | -0.18 |
| 7 | 48.83 | +0.18 |
| 8 | 48.25 | -0.58 |
| 9 | 49.11 | +.086 |
| 10 | 49.41 | +0.30 |

At the end of 4 years ** from 1964 - 1968 the timing has increased by 0.02 second from 52.01 to 52.03 second.

At the end of second Olympic the reduction in timing is 0.95 second from 52.03 to 51.08. From the table above we cannot find a definite relationship between the number of years and change in timing. But the reduction is fairly steady except in the first 7th year, 9th year and 10th year.

The mean of the value is

Le us assume that the timings in 400m race of Olympic reduced at the rate of 0.26per Olympic.

Mathematical description :

We have assumed that the timings reduces at the rate of 0.26 second per Olympic.

So, the reduction in timings in the first Olympic = 52.01 - 0.26

Reduction in the second Olympic = 52.01 - 0.26 - 0.26 = 52.01 - 2 × 0.26

Reduction in the third Olympic = 52.01 - 0.26 - 0.26 - 0.26 = 52.01 - 3 × 0.26

So, the reduction in the 11th Olympic = 52.01 - 11 × 0.26

Now, we have to estimate the timings in the next (11th) Olympic i.e., 2008

But the timings in the nth Olympic will be = 52.01 - 0.26 n ....(1)

Step 2. Solution : Substituting n = 11, in (1), we get 52.01 - 0.26 × 11 = 52.01 - 2.86 = 49.15

Step 3. Interpretation : The timings for 400m race in the next Olympic i.e., (2008) is estimated as 49.15sec.

Step 4. Validation : Let us check if formula (1) is in agreement with the reality. Let us find the values for the years we already know using formula (1) and compare it with known values by finding the difference.