#### To determine

**To find:** The rate of population growth in 1920 and in 1980 by averaging the slopes of two secant lines.

#### Answer

The rate of population growth in 1920 and in 1980 by averaging the slopes of two secant lines is 16million/year and 78.5million/year.

#### Explanation

**Given:**

The population of the world P(t) corresponds to the year 1900 is shown Table 1.

t | Population (millions) |

0 | 1650 |

10 | 1750 |

20 | 1860 |

30 | 2070 |

40 | 2300 |

50 | 2560 |

60 | 3040 |

70 | 3710 |

80 | 4450 |

90 | 5280 |

100 | 6080 |

110 | 6870 |

Table 1

Calculate the rate of population growth in 1920.

Calculate the slope m1.

m1=y2−y1x2−x1

Substitute 1860 for y2, 1750 for the expression y1, 1920 for x2 and 1910 for x1 in the above equation.

m1=1860−17501920−1910=11010=11

Calculate the slope m2.

m2=y2−y1x2−x1

Substitute 2070 for y2, 1860 for y1, 1930 x2 and 1920 for x1 in the above equation.

m2=2070−18601930−1920=21010=21

Obtain the average slope as follows.

m=m1+m22

Substitute 11 for m1 and 21 for m2 in the above equation.

m=11+212=322=16 millions/year

Thus, the rate of population growth in 1920 is 16 millions/year.

Calculate the rate of population growth in 1980.

Calculate the slope m1.

m1=y2−y1x2−x1

Substitute 4450 for y2, 3710 for y1, 1980 for the expression x2 and 1970 for x1 in the above equation.

m1=4450−37101980−1970=740110=74

Calculate the slope m2 using the formula.

m1=y2−y1x2−x1

Substitute 5280 for y2, 4450 for y1, 1990 for x2 and 1980 for x1 in the above equation.

m2=5280−44501990−1980=83010=83

Obtain the average slope as follows.

m=m1+m22

Substitute 74 for the slope m1 and 83 for the slope in the above equation.

m=m1+m22=74+832=1572=78.5 millions/year

Thus, the rate of population growth in 1980 is 78.5 millions/year.

#### To determine

**To find:** To find the cubic function (a third-degree polynomial) that models the given data.

#### Answer

A third-degree polynomial for the given model is P(t)=−0.0002849003t3+0.522433t2−6.39564t+1720.586.

#### Explanation

Calculate the third degree polynomial for the model.

P(t)=at3+bt2+ct+d (1)

Plot the graph between t and population using the table (1).

Refer the figure (1).

The third-degree polynomial for the curve is

P(t)=−0.0002849003t3+0.522433t2−6.39564t+1720.586 (2)

Equate the equation (1) with equation (2).

at3+bt2+ct+d=−0.0002849003t3+0.522433t2−6.39564t+1720.586

Compare the above equation.

a=−0.0002849003, b=0.522433, c=−6.39564 and d=1720.586.

Therefore, the third-degree polynomial for the given model is P(t)=−0.0002849003t3+0.522433t2−6.39564t+1720.586.

#### To determine

**To find:** To find the model for the rate of population growth.

#### Answer

Rate of population growth for the model is P'(t)=3at2+2bt+c(in millions of people per year).

#### Explanation

Calculate a model for the rate of population growth.

Differentiate equation (1) with respect to t.

P'(t)=3at2+2bt+c (3)

Thus, the rate of population growth for the model is P'(t)=3at2+2bt+c(in millions of people per year).

#### To determine

**To find:** The rates of growth in 1920 and 1980 and compare your estimates in part (a).

#### Answer

The growth rate in 1920 and 1980 is 14.16millions/year and 76.71 millions/year.

#### Explanation

Estimate the rate of growth in 1920.

Substitute 20 for t in the equation (2).

P'(20)=3a(20)2+2b(20)+c=1200a+40b+c

Substitute −0.0002849003 for a, 0.522433 for b and −6.39564 for c in the above equation.

P'(20)=1200(−0.0002849003)+40(0.522433)−6.39564=14.16millions/year

Thus, the growth rate in 1920 is 14.16millions/year.

From part (a), the rate of population growth in 1920 is 16 millions/year, which is greater than the obtained result.

Estimate the rate of growth in 1980.

Substitute 80 for t in the equation (2).

P'(80)=3a(80)2+2b(80)+c=19200a+160b+c

Substitute −0.0002849003 for a, 0.522433 for b and −6.39564 for c in the above equation.

P'(80)=19200(−0.0002849003)+160(0.522433)−6.39564=−5.4700+83.59−6.39564=71.72 millions/year

Thus, the growth rate in 1980 is 76.71 millions/year.

From part (a), the rate of population growth in 1980 is 78.5 millions/year, which is greater than the obtained result.

#### To determine

**To find:** The rate of population growth in 1985.

#### Answer

Rate of population growth is 76.89million/yr.

#### Explanation

Calculate the rate of population growth in 1985 as follows.

Differentiate equation (2) with respect to t.

P'(t)=−0.0007647009t2+1.044866t−6.39564

Substitute 85 for t in the above equation.

P'(85)=−0.0007647009(85)2+1.044866(85)−6.39564=76.89million/yr

Thus, the rate of population growth is 76.89million/yr.