1. The U.S. Constitution requires a census every 10 years. The census data for 1790-2010 are given in the table.

Year t	PopulationP (in millions)		Year t	PopulationP (in millions)		Year t	PopulationP (in millions)
1790	3.9		1870	38.6		1950	151.3
1800	5.3	1880	50.2	1960	179.3
1810	7.2	1890	63.0	1970	203.3
1820	9.6	1900	76.2	1980	226.5
1830	12.9	1910	92.2	1990	248.7
1840	17.1	1920	106.0	2000	281.4
1850	23.2	1930	123.2	2010	308.7
1860	31.4	1940	132.2

(b) Use a calculator to find an exponential modelP(t) =Cb^tfor the data. (WriteCin scientific notation. Round the decimal value ofCand the value ofbto seven decimal places.) P(t) =

(c) Use your model to predict the population at the2040census. (Round your answer to one decimal place.) million (d) Use your model to estimate the population in1985. (Round your answer to one decimal place.) million

2. A student is trying to determine the half-life of radioactive iodine-131. He measures the amount of iodine-131 in a sample solution every 8 hours. His data are shown in the table below.

Time t (h)	Amount of 131I A (g)
0	4.80
8	4.66
16	4.51
24	4.39
32	4.29
40	4.14
48	4.04

(b) Use a calculator to find an exponential model. (Round all numerical values to three decimal places.) A(t) =

(c) Use your model to find the half-life of iodine-131. (Round your answer to one decimal place.) hr

3. As sunlight passes through the waters of lakes and oceans, the light is absorbed, and the deeper it penetrates, the more its intensity diminishes. The light intensityIat depthxis given by the Beer-Lambert Law:

I=I₀e^kx

whereI₀is the light intensity at the surface andkis a constant that depends on the murkiness of the water. A biologist uses a photometer to investigate light penetration in a northern lake, obtaining the data in the table.

Depth (ft)	Light intensity (lm)		Depth (ft)	Light intensity (lm)
5	12.4		25	1.7
10	8.0	30	1.0
15	4.5	35	0.5
20	2.6	40	0.3

Light intensity decreases exponentially with depth.

(a) Use a graphing calculator to find an exponential function of the form given by the Beer-Lambert Law to model these data. What is the light intensityI₀at the surface on this day, and what is the "murkiness" constantkfor this lake? [Hint:If your calculator gives you a function of the form

I=ab^x,

convert this to the form you want using the identities

b^x=e^ln(bx)=e^xln(b).]

(Round all numerical values to four decimal places.)

I=

(c) If the light intensity drops below 0.15 lumen (lm), a certain species of algae can't survive because photosynthesis is impossible. Use your model from part (a) to determine the depth below which there is insufficient light to support this algae. (Round your answer to two decimal places.) ft

4. Data points

(x, y)

are shown in the table.

x	2	4	6	8	10	12	14	16
y	0.07	0.12	0.19	0.26	0.35	0.52	0.72	1.06

(d) Find an appropriate function to model the data. (Round all numerical values to five decimal places.)

y =

5. A graphing calculator is recommended. The relative growth rate of world population has been decreasing steadily in recent years. On the basis of this, some population models predict that world population will eventually stabilize at a level that the planet can support. One such logistic model is

P(t) =

74.4

6.2 + 6.4e^0.02t

where t = 0 is the year 2000 and population is measured in billions.(a) What world population does this model predict for the year 2220? For 2320? (Round your answers to two decimal places.)

220 billion

2320. billion

(c) According to this model, what size does the world population seem to approach as time goes on? billion

6. A small lake is stocked with a certain species of fish. The fish population is modeled by the function

P =

14

1 + 4e^0.9t

where P is the number of fish in thousands and t is measured in years since the lake was stocked.

(a) Find the fish population after 2 years. (Round your answer to the nearest whole fish.) fish (b) After how many years will the fish population reach 7000 fish? (Round your answer to two decimal places.) yr

7. The table and scatter plot give the population of black flies in a closed laboratory container over an 18-day period.

Time (days)	Number of flies
0	10
2	25
4	66
6	144
8	262
10	374
12	446
16	492
18	498

(a) Use theLogisticcommand on your calculator to find a logistic model for these data. (Round all numeric values to four decimal places.)

N(t) =

(b) Use the model to estimate the time when there were400flies in the container. (Round your answer to two decimal places.) days