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Question 1014454: In 1995, there were 41,973 shopping centers in a certain country. In 2005, there were 48,168.
a. Write an equation expressing the number y of shopping centers in terms of the number x of years after 1995.
Write in y=mx+b form
b. When will the number of shopping centers reach 80,000?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! in 1995, there were 41,973 shopping centers.
in 2005, there were 48,168.
slope intercept form of the equation of a straight line is:
y = mx + b
m is the slope
b is the y-intercept.
if you let x equal the number of years after 1995, then you get:
when x = 0, the year is 1995 because 1995 + 0 = 1995
when x = 1, the year is 1996 because 1995 + 1 = 1996
when x = 10, the year is 2005 because 1995 + 10 = 2005.
when x = 0, y = 41,973
when x = 10, y = 48,168
you need to find the slope of your equation.
the slope is equal to (y2-y1)/(x2-x1)
set (x1,y1) = (0,41973)
set (x2,y2) = (10,48168)
(x1,y1) and (x2,y2) are points on the line of your equation.
slope = (y2-y1)/(x2-x1) becomes slope = (48168-41973 / (10-0) which becomes slope = 6195/10 which becomes slope = 619.5
your slope is 619.5.
you need to find the y-intercept.
your slope intercept form of the equation is now y = 619.5 + b because m is the slope and the slope is 619.5.
you can use any point on your equation to solve for b.
when x = 0, y = 41973.
use the point (x,y) = (0,41973).
replace x with 0 and y with 41973 and the equation becomes 41973 = 619.5*0 + b
solve for b to get b = 41973.
since the y-intercept is the value of y when x is equal to 0, this makes sense because we already knew that the value of y when x = 0 is 41973.
your equation now becomes y = 619.5 * x + 41973.
if we did this correctly, when x = 10, y should be equal to 48168.
10*619.5 = 6195 + 41973 = 48168.
we did it correctly.
to solve for when the number of shopping centers reaches 80,000, replace y in that equation with 80,000 and solve for x.
you will get 80,000 = 619.5*x + 41973
subtract 41973 from both sides of this equation to get 80,000 - 41,973 = 619.5 * x
simplify to get 38,027 = 619.5 * x
divide both sides of this equation b y 619.5 to get x = 61.38337369
you will reach 80,000 centers sometime in year 61.
let's try this out and see if it's correct.
y = 619.5 * x + 41973.
when x = 61, y = 79,762.5
when x = 62, y = 80,382
the solution is correct.
at the beginning of year 61, you had less than 80,000 and at the beginning of year 62, you had more than 80,000, so sometime in year 61 you reached 80,000.
year 61 is equal to 1995 + 61 = the year 2056
2056 is 61 years after 1995.
the graph of your equation with the important intersection points shown is below.
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