Question

question 7
in which decade (10 - year interval) did the population grow the fastest?
explain how you know.
from 1950 to 1960. the graph has the steepest slope between the two points
on the graph (1950, 45) and (1960, 43).
from 1910 and 1920. the graph has the steepest slope between the two points
on the graph (1910, 22) and (1920, 32).
from 1900 to 1910. the graph has the steepest slope between the two points
on the graph (1990, 20) and (1910, 22).
from 1980 to 2000. the graph has the steepest slope between the two points
(1980, 30) and (2000, 26).

Explanation:

Step1: Calculate the slope for each option

The formula for slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
For the option "From 1950 to 1960" with points \((1950,45)\) and \((1960,43)\):
\(m_1=\frac{43 - 45}{1960 - 1950}=\frac{- 2}{10}=-0.2\)
For the option "From 1910 and 1920" with points \((1910,22)\) and \((1920,32)\):
\(m_2=\frac{32 - 22}{1920 - 1910}=\frac{10}{10}=1\)
For the option "From 1900 to 1910" with points \((1900,20)\) and \((1910,22)\):
\(m_3=\frac{22 - 20}{1910 - 1900}=\frac{2}{10}=0.2\)
For the option "From 1980 to 2000" with points \((1980,30)\) and \((2000,26)\):
\(m_4=\frac{26 - 30}{2000 - 1980}=\frac{-4}{20}=-0.2\)

Step2: Compare the slopes

We are looking for the fastest population growth (positive slope, since growth implies an increase in population). Among \(m_1=-0.2\), \(m_2 = 1\), \(m_3=0.2\), \(m_4=-0.2\), the largest positive slope is \(m_2\).

Answer:

From 1910 and 1920. The graph has the steepest slope between the two points on the graph \((1910,22)\) and \((1920,32)\) as its slope \(m = 1\) (calculated using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\)) which is the largest positive slope among all the given - decade intervals.