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.
If
N(t)=mt+b
such that N(1950)=A and N(2000)=B, then m= and 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.)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.
If
N(t)=mt+b
such that N(1950)=A and N(2000)=B, then m= and 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.)
N(1975) = 4,317,183,977
To find the values of m and b in the equation N(t) = mt + b, we need to use the given information that N(1950) = A and N(2000) = B.
Plug in the values of t = 1950 and N(t) = A into the equation N(t) = mt + b:
A = 1950m + b
Similarly, plug in the values of t = 2000 and N(t) = B into the equation N(t) = mt + b:
B = 2000m + b
Now, we have a system of two equations with two unknowns (m and b). We can solve this system of equations to find the values of m and b.
Subtract the first equation from the second equation to eliminate b:
B - A = (2000m + b) - (1950m + b)
B - A = 2000m - 1950m
Simplify:
B - A = 50m
Now divide both sides of the equation by 50:
(B - A)/50 = m
Next, substitute the value of m back into one of the original equations to solve for b. Let's use the first equation:
A = 1950m + b
Plug in the value of m:
A = 1950((B - A)/50) + b
Multiply both sides by 50 to eliminate the fraction:
50A = 1950(B - A) + 50b
Multiply and distribute:
50A = 1950B - 1950A + 50b
Simplify:
2450A = 1950B + 50b
Now divide both sides by 50:
49A = 39B + b
Subtract b from both sides:
49A - b = 39B
Substitute the value of (B - A)/50 for m:
49A - b = 39B
Solve the equation above for b:
b = 39B - 49A
Now that we have the values of m and b, we can use the equation N(t) = mt + b to estimate the population in 1975. Plug in t = 1975 into the equation:
N(1975) = m(1975) + b
Substitute the values of m and b and calculate the population in 1975.