Question

A graphing calculator is recommended. The table gives the population of the world $P(t)$, in millions, where $t$ is measured in years and $t = 0$ corresponds to the year 1900. egin{tabular}{|c|c|c|c|} hline $t$ & Population \ (in millions) & $t$ & Population \ (in millions) \ hline 0 & 1650 & 60 & 3040 \ 10 & 1750 & 70 & 3710 \ 20 & 1860 & 80 & 4450 \ 30 & 2070 & 90 & 5280 \ 40 & 2300 & 100 & 6080 \ 50 & 2560 & 110 & 6870 \ hline end{tabular} (a) Estimate the rate of population growth in 1920 and in 1980 by averaging the slopes of two secant lines. growth in 1920 16 million/year growth in 1980 78.5 million/year (b) Use a graphing device to find a cubic function (a third-degree polynomial) that models the data. (Let $t$ be the year. Round all numerical values to five decimal places.) $P(t) = $ (c) Use your model in part (b) to find a model for the rate of population growth. Then, estimate the rates of growth in 1920 and 1980. Compare with your estimates in part (a). (Round your answers to two decimal places.) $P'(20) = $ million/year $P'(80) = $ million/year

          A graphing calculator is recommended.
The table gives the population of the world $P(t)$, in millions, where $t$ is measured in years and $t = 0$ corresponds to the year 1900.
egin{tabular}{|c|c|c|c|}
hline
$t$ & Population \
(in millions) & $t$ & Population \
(in millions) \
hline
0 & 1650 & 60 & 3040 \
10 & 1750 & 70 & 3710 \
20 & 1860 & 80 & 4450 \
30 & 2070 & 90 & 5280 \
40 & 2300 & 100 & 6080 \
50 & 2560 & 110 & 6870 \
hline
end{tabular}
(a) Estimate the rate of population growth in 1920 and in 1980 by averaging the slopes of two secant lines.
growth in 1920 16 million/year
growth in 1980 78.5 million/year
(b) Use a graphing device to find a cubic function (a third-degree polynomial) that models the data. (Let $t$ be the year. Round all numerical values to five decimal places.)
$P(t) = $ 
(c) Use your model in part (b) to find a model for the rate of population growth. Then, estimate the rates of growth in 1920 and 1980. Compare with your estimates in part (a).
(Round your answers to two decimal places.)
$P'(20) = $ million/year
$P'(80) = $ million/year

Show more…

A graphing calculator is recommended.
The table gives the population of the world P(t), in millions, where t is measured in years and t = 0 corresponds to the year 1900.
egintabular|c|c|c|c|
hline
t Population (in millions) t Population (in millions) hline
0 1650 60 3040 10 1750 70 3710 20 1860 80 4450 30 2070 90 5280 40 2300 100 6080 50 2560 110 6870 hline
endtabular
(a) Estimate the rate of population growth in 1920 and in 1980 by averaging the slopes of two secant lines.
growth in 1920 16 million/year
growth in 1980 78.5 million/year
(b) Use a graphing device to find a cubic function (a third-degree polynomial) that models the data. (Let t be the year. Round all numerical values to five decimal places.)
P(t) =
(c) Use your model in part (b) to find a model for the rate of population growth. Then, estimate the rates of growth in 1920 and 1980. Compare with your estimates in part (a).
(Round your answers to two decimal places.)
P'(20) = million/year
P'(80) = million/year

Added by Corey T.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert William Semus

03/22/2023

Step 1

First, we need to organize the data in a table with years and population: | Year (t) | Population (P) | |----------|----------------| | 0 | 650 | | 20 | 1750 | | 40 | 2300 | | 60 | 3040 | | 80 Show more…

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A graphing calculator is recommended. The table gives the population of the world P(t), in millions, where t is measured in years and t = 0 corresponds to the year 1900. | t | Population (in millions) | t | Population (in millions) | |---|---|---|---| | 0 | 1650 | 60 | 3040 | | 10 | 1750 | 70 | 3710 | | 20 | 1860 | 80 | 4450 | | 30 | 2070 | 90 | 5280 | | 40 | 2300 | 100 | 6080 | | 50 | 2560 | 110 | 6870 | (a) Estimate the rate of population growth in 1920 and in 1980 by averaging the slopes of two secant lines. growth in 1920 16 million/year growth in 1980 78.5 million/year (b) Use a graphing device to find a cubic function (a third-degree polynomial) that models the data. (Let t be the year. Round all numerical values to five decimal places.) P(t) = (c) Use your model in part (b) to find a model for the rate of population growth. Then, estimate the rates of growth in 1920 and 1980. Compare with your estimates in part (a). (Round your answers to two decimal places.) P'(20) = million/year P'(80) = million/year

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00:01 So in this question, they say this table gives the population of the world p of t in millions, where t is measured in years and t equals zero corresponds to the year in 1900.

00:11 In part a, they said estimate the rate of population growth in 1920 and 1980 by averaging the slopes of the two seacan lines.

00:20 And it looks like you have that correct.

00:23 So now we're going to move to part b, where they say use a graphing device to find a cubic function that models the data.

00:29 I'm going to let tb the year and i'm going to round all my values to five decimal places.

00:34 So i'm going to head to my calculator and i'm going to go into the stat menu and i'm going to edit.

00:41 And in l1, i'm going to put these xs, which go by tens from zero down to 110.

00:54 So 40, 50, 60, 70, 70, 80.

01:05 90, 100, 110.

01:11 And now i'm going to go over to l2.

01:13 What i'm going to put in my ys.

01:16 So my ys are 1650, 1750, followed by 1860, followed by 2070, and then it was 2300, and then it's 2560, and then it's 3040, and then it's 3040, 3710, and then it's 4 ,450, and then it was 5 ,280, followed by 6 ,080, and then finally, 6870.

02:17 Now i'm going to find a cubic regression.

02:19 So i'm going to go to stat, calculate.

02:22 Number six, a cubic regression, and let's see what we get for this cubic regression.

02:32 I get.

02:34 So my a is negative 2 .84900.

02:41 So what is happening there? that is going to be negative.

02:50 0 .00028, i believe, because i've got 2 .8 times 10 to the negative 4th...

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