Question

In 19951995​, there were 41 comma 43141,431 shopping centers in a certain country. in 20052005​, there were 48 comma 72948,729. ​(a) write an equation expressing the number y of shopping centers in terms of the number x of years after 19951995. ​(b) when will the number of shopping centers reach 80 comma 00080,000​?

Answer 1

(a) y = 729.8x + 41431
(b) 80,000 shopping centers in about 2048.

(a) Since we only have 2 data points, let's create the equation of a line using slope intercept form where y = number of shopping centers and x = number of years since 1995. So the general form of the equation will be:
y = ax + b

Let's first calculate a, which is the difference in y divided by the difference in x, so
(48729 - 41431)/(2005-1995) = 7298/10 = 729.8

Now we have the equation
y = 729.8x + b

Substitute the values for a known pair and solve for b. I'll use the pair (0, 41431). The value 0 represents 0 years after 1995. So
y = 729.8x + b
41431 = 729.8*0 + b
41431 = 0 + b
41431 = b

So the desired equation is:
y = 729.8x + 41431

(b) To answer this, substitute 80000 for y and then solve for x. So:
y = 729.8x + 41431
80000 = 729.8x + 41431
38569 = 729.8x
52.84872568 = x
So the number of shopping centers will reach 80000 52.8 years after 1995, or about 2048.

Answer 2

The first thing you should do for this case is to use the following table to perform the equation of a line:
y x
41,431 1995
48,729 2005
We have then that the line that best fits this data is
y = 729.8x - 1E + 06
Then, to know in what year the number of shopping centers reaches 80,000 we must replace this number in the equation of the line and clear x:
80000 = 729.8x - 1E + 06
Clearing x
x = (80000 + 1E + 06) / (729.8) = 1479.857495
nearest whole number
1480
This means that after 1480 years, 80000 shopping centers are reached.
Equivalently, this amount is reached in the year:
1480 + 1995 = 3475
In the year 3475
answer
(a) y = 729.8x - 1E + 06
(b) In the year 3475