The number of visitors at a Web site

over several days is shown in the table
at the right. What is an equation that models the data?

This is the given data:

Days: 1, 2, 3, 4, 5
Visitors: 52, 197, 447, 805, 1270

How do you write the equation?

Just noticed a typo (sticking 3 key)

<taking the first 3 I got:
a = 5.5 , b=-12.5 , c = 12>
should say:

taking the first 3 I got:
a = 53.5 , b=-12.5 , c = 12

btw , I did not show how I solved those 3 equations in 3 unknowns, I used a webpage for that
http://www.1728.org/unknwn3.htm

I did the usual analysis and found the first differences to be

145, 250, 358, and 465
and the second differences to be
105, 108, and 107
which appear to be relatively constant at appr 106, so I suspect a quadratic.
let N = ad^2 + bd + c, where d is the number of days, and N is the number of visitors.
we get:
52 = a + b + c
197 = 4a + 2b + c
447 = 9a + 3b + c
805 = 16a + 4b + c
1270 = 25a + 5b + c
Since we have more than 3 equations in only 3 unknowns, and our second differences were really not constant, our solutions could only be approximate.

taking the first 3 I got:
a = 5.5 , b=-12.5 , c = 12
taking the 1st, 3rd and 5th, I got
a = 53.5, b= -16.5, c = 15

approximating:
N = 53.5n^2 - 16.5n + 15

I know I get exact values for n = 1, 3, 5
how about n = 2 and n=4
N(2) = 196 instead of 197 , not bad
N(4) = 805 , right on!

I would say my equation is pretty good

To write an equation that models the data, we need to identify the pattern or relationship between the number of days and the number of visitors.

From the given data, let's observe the change in the number of visitors over consecutive days:

Number of visitors on Day 2 = Number of visitors on Day 1 + Increase in visitors from Day 1 to Day 2
Number of visitors on Day 3 = Number of visitors on Day 2 + Increase in visitors from Day 2 to Day 3
Number of visitors on Day 4 = Number of visitors on Day 3 + Increase in visitors from Day 3 to Day 4
Number of visitors on Day 5 = Number of visitors on Day 4 + Increase in visitors from Day 4 to Day 5

To calculate the increase in visitors from one day to another, we can use the concept of a common difference. The common difference represents the constant rate at which the number of visitors is increasing each day.

Let's calculate the common difference:

Common difference from Day 1 to Day 2 = 197 - 52 = 145
Common difference from Day 2 to Day 3 = 447 - 197 = 250
Common difference from Day 3 to Day 4 = 805 - 447 = 358
Common difference from Day 4 to Day 5 = 1270 - 805 = 465

We can observe that the common difference changes slightly in each progression, so it is not a linear progression. Instead, it appears to be increasing or growing.

To model this growth, we can use quadratic or exponential equations. Let's try using an exponential equation:

y = a * (b)^x

Where:
y = Number of visitors
x = Number of days
a = Initial value (the number of visitors on Day 1)
b = Growth factor or rate of increase

Using the first data point (Day 1: 52 visitors) as the initial value (a), let's find the growth factor (b) by using a subsequent data point:

197 = 52 * (b)^1

Solving for b, we get:

b ≈ √(197/52) ≈ 1.4525

Therefore, we can write the equation that models the data as:

y ≈ 52 * (1.4525)^x

Thanks!