The next three problems explore some techniques of data analysis. According to the US Bureau of the Census, the world population in the year 1950 was A=2555360972 , and in 2000 it was B=6079006982. We'll use A and B so we don't have to keep writing those large and idiosyncratic numbers. We usually use y and x in the equation of a line, but in this and the following problem let's use N and t instead. t stands for time and N for the size of the population.

Question

If N(t)=mt+b ...such that N(1950)=A and N(2000)=B, then m=_________ and b=__________. Suppose you want to estimate the population in 1975. To that end you compute N(1975) = _________. (Round your answers to the nearest integer. The process illustrated in this problem is called linear interpolation.)

Answer 1

To find the values of m and b, we can use the given information that N(1950) = A and N(2000) = B.

First, let's find the slope, m:

m = (B - A) / (2000 - 1950)
m = (6079006982 - 2555360972) / (2000 - 1950)
m = 3523646010 / 50
m = 70472920.2

Next, let's find the y-intercept, b:

We know that N(1950) = A, so we can substitute these values into the equation:

A = m * 1950 + b
2555360972 = (70472920.2 * 1950) + b
2555360972 = 137378277790 + b
b = 2555360972 - 137378277790
b = -134822917818

So, m = 70472920.2 and b = -134822917818.

Now, let's find the estimated population in 1975 using linear interpolation:

N(1975) = m * 1975 + b
N(1975) = (70472920.2 * 1975) + (-134822917818)
N(1975) = 139199115975 - 134822917818
N(1975) = 4371194157

Therefore, the estimated population in 1975 is 4371194157.