Question

3. The number of visitors to a small pumpkin patch that opened in 2003 increased each year as shown in the table. Predict the number of visitors in 2015, assuming the increase continues at this same rate.

Year 2003 2004 2005 2006 2007 2008
Visitors 310 332 355 380 406 434

Answer 1

The predicted number of visitors in 2015 is 670.

Answer 2

The predicted number of visitors in 2015 is 605.

Explanation:

Since, the difference in number of visitors in consecutive year is approximately constant.

Thus, the function that shows the given situation can be linear.

Let x represents the number of years and y represents the visitors.

Now, suppose the estimation is started from 2003,

Hence, the table will be,

Year (x) 0 1 2 3 4 5

Visitors (y) 310 332 355 380 406 434

By the above table,

`\sum x = 15`

`\sum y = 2217`

`\sum x^2 = 55`

`\sum xy = 5976`

Since, the equation of a linear function is,

y = b + ax

Where,

`a=(\sum y \sum x^2 - \sum x \sum xy)/(n(\sum x^2)-(\sum x)^2)`

And,

`b=(n(\sum xy) - \sum x \sumy)/(n(\sum x^2)-(\sum x)^2)`

Where, n is the number of observation,

Here, n = 6,

By putting the values,

We get, a = 24.77142857 ≈ 24.77 and b = 307.5714286≈ 307.57

Thus, the equation that shows the given table is,

y = 24.77 x + 307.57

For 2015, x = 12,

⇒ y = 24.77 × 12 + 307.57 = 604.81 ≈ 605.

Hence, the predicted number of visitors = 605.