Explain why it is reasonable that the graph is increasing

Question

1. The number N (in thousands) of inhabitants in the state of Gamma is a function of time t, measured in years since the state was settled. The formula is

N= 6.2 / (0.1 + 0.5^t)

(a) Make a graph of N versus t and draw that graph on your paper. Include times up to 10 years, and be sure to indicate the window you use.

(b) Explain in practical terms what N(5) means, and then calculate it.

(c) As you can see from the graph in Part (a), this population growth is logistic. Use your graph or a table of values to answer the following questions.

i.  What is the carrying capacity for the population in this environment? (Note: The carrying capacity does not equal the numerator, 6.2, in the formula for N, since this formula is not in standard form for a logistic function. Instead, use your graph or a table to find the carrying capacity.)

ii. At what population size N is the population growing the fastest? Explain how you got your answer.

iii. What portion of the graph is concave up? Explain in practical terms what this means.

2. In an attempt to predict the growth of the population of the U.S., biologists studied census records from 1790 through 1940. They developed the logistic formula

N= 184/ (1+ 66.7e^0.03t)

Here N is the U.S. population in millions and t is time (measured in years since 1780).

(a) Make a graph of N versus t and draw that graph on your paper. Include times up to 250 years (corresponding to dates up to 2030).

(b) Use your graph or a table of values to determine the carrying capacity for the population. Does this make the model look accurate?

(c) According to this model, on what date was the U.S. population growing the fastest?

(d) Suppose another biologist has proposed an alternative logistic formula to model the U.S. population. Under this model, the population level at which the popu­lation grows the fastest is 150 million. What would the carrying capacity for the population be under this new model?

3. The tables below show linear, exponential, or power data. Determine which is which, explain your reasoning, and write a formula for the function in each case.

Table A

t	0	1	2	3	4	5
f (t)	6.7	7.77	9.02	10.46	12.13	14.07

Table B

t	1	2	3	4	5
g(t)	7.7	17.2	27.54	38.45	49.81

Table C

t	0	1	2	3	4	5
h(t)	5.8	7.53	9.26	10.99	12.72	14.45

4. The tables below show data modeled by a linear function, an exponential function, and a power function.

(a)

t	1	2	3	4
f (t)	5	5.5	6.05	6.66

i.  What type of function is f: linear, exponential, or power? Why?

ii. Give a formula for f.

(b)

t	1	2	3	4
g(t)	5	5.5	5.82	6.05

i. What type of function is g: linear, exponential, or power? Why?

ii. Give a formula for g.

(c)

t	1	2	3	4
h(t)	5	5.5	6	6.5

i.  What type of function is h: linear, exponential, or power? Why?

ii. Give a formula for h.

5. (a) The tax you owe (in dollars) is a linear function of your taxable income (in dollars), and the slope of this function is 0.12. If your income increases by $150, how much more tax will you owe?

(b) The circulation of a certain magazine is an exponential function of time, with yearly growth factor 1.05. By what factor will the circulation increase over a decade?

(c) The speed S at which a bird can fly is a power function of its length L, and the power is k = 0.3. If one bird is twice as long as another, how much faster can it fly?

6. The weight W in ounces for a certain species of lizard is a power function of its length L in inches.

(a) If one lizard is twice as long as a second, then the first weighs 3 times as much as the second. What is the value of the power k in the relationship between W and L?

(b) If one lizard is 4 times as long as a second, how do their weights compare?

(c) If one lizard weighs twice as much as a second, how do their lengths compare?

7. For similarly shaped objects, terminal velocity T is proportional to the square root of length L.

(a) How does the terminal velocity of a 4-foot monkey compare with that of a 3-foot monkey?

(b) The terminal velocity of a 4-foot monkey is twice that of a certain smaller mammal with a similar shape. How long is this smaller mammal?

8. For a satellite orbiting the earth, the distance D from the center of the Earth is a power function of the period P. Let k denote the power.

(a) If the period of one satellite is twice that of another, its distance from the center of the Earth is 1.59 times larger. Find the value of the power, k.

(b) If the period of one satellite is 3 times that of a second, how do their distances from the center of the Earth compare?

(c) If the distance from the center of the Earth of one satellite is twice that of a second, how do their periods compare?

9. The table below shows the relationship between the length L in centimeters and weight W in grams of a certain species of fish.

L	31	33	35	37	39
W	250	308	363	420	520

(a) Find a formula that models W as a power function of L.

(b) If one fish is twice as long as another, how do their weights compare?

(c) If one fish is twice as heavy as another, how do their lengths compare?

10. The following table gives the power P in watts generated by a windmill with winds blowing v miles per hour.

v	5	10	15	20	25
P	1.8	15	50.7	120	234

(a) Plot the graph of In P versus In v. Is it reasonable to model P as a power function of v? Explain your reasoning.

(b) Find a model of P as a power function of v.

(c) What power is generated by 35 mile per hour winds?

(d) How fast must the wind blow in order to generate 41 watts of power?

(e) If wind speed increases by a factor of 3, how much more power is generated?

11. The volume V of a sphere (ball) of radius r is given by

V = (4/3) πr³.

The surface area S is given by

S = 4 πr².

(a) Show that the radius can be expressed as a function of surface area by the formula

r= √(S /4π)

(b) Use function composition to find a formula expressing volume as a function of surface area.

12. The weight W in ounces of a certain small mammal is proportional to the cube of its length L in inches.

(a) Express W as a function of L using c as the constant of proportionality.

(b) The length in inches of the animal depends on its age t in years. The relationship is as follows:

L = 8(1- e^-t).

Use function composition to find a formula expressing weight as a function of time.

(c) It is found that a 3-year-old animal weighs 17 ounces. Find the value of c.

 

13. For a certain species of fish it is found that length L is a function of age t. The maximum length of this species is 13 inches. The youngest fish of this species are 0.5 inch long. If D denotes the difference between maximum length and current length, then

L = Limiting value - D.

(a) What is the limiting value?

(b) It is found that D is an exponential function of t. What is the initial value of D?

(c) It is found that a 3-year-old fish of this species is 8 inches long. Find a formula for D in terms of t.

(d) Find a formula for L in terms of t.

(e) It is found that the weight W in ounces of this species of fish is given by W = 0.01L³. Use function composition to find a formula expressing weight as a func­tion of age.

14. You begin a voter registration drive in a town with a voting-age population of 40,000. Let D denote the difference between the voting-age population and voters registered. The total number R of registrations is given by

R= Limiting value - D.

(a) What is the limiting value of R?

(b) It is found that D is an exponential function of the number t of months since the drive began. There were 28,000 registered voters when the drive began. What is the initial value of D?

(c) After one month, you find that a total of 33,000 voters are registered. Find a for­mula for D in terms of t.

(d) Find a formula for R in terms of t

15. One of the two tables below shows data that can be modeled by a quadratic function, and the other shows data that cannot be modeled by a quadratic function. Identify which is which, and find a model for the quadratic data.

Table A

x	0	1	2	3	4
f (x)	2	0	4	16	32

Table B

x	0	1	2	3	4
g(x)	2	0	4	14	30

16. A cannonball is fired from a cannon. Its height h in feet is measured d feet downrange and is recorded in the table below.

d	0	300	500	800	1000
h	0	181	236	220	144

(a) We know that the cannonball should follow the path of a parabola. Use quadratic regression to find a parabola that is approximately followed by the cannonball.

(b) How far downrange will the cannonball strike the ground?

(c) If the cannon has a slope of inclination s and the initial velocity is vo feet per second, then the path followed by the cannonball is the graph of

-16 {(1 + s²) / v₀²} x² +sx

Based on your answer to Part (a), what is the slope of inclination of the can­non?

i.  Based on your answer to Part (a), what is the initial velocity of the cannonball?

17. A manufacturer has recorded the profit P (in dollars) when there is a monthly advertising expenditure of A dollars. The data is recorded in the following table.

A	200	500	800	1100	1300
P	8030	14,480	14,630	8480	860

(a) Find a quadratic model for profit as a function of advertising expenditure.

(b) What advertising expenditure gives a maximum profit, and what is that profit?

18. A manufacturer has determined that profit P (in dollars) is a quadratic function of dollars per month a spent on advertising. The relationship is given by

P = 1 + 50a - 0.1a²

if a is at most 500 dollars per month.

(a) How much should be spent on advertising if profit is to be a maximum?

(b) How much should be spent on advertising if a profit of $5000 is desired?

(c) The manufacturer is spending $350 per month on advertising. Should the manu­facturer increase or decrease that amount?

19. Consider the quadratic x² + kx + k = 0.

(a) Solve the equation using the quadratic formula.

(b) What condition on k will assure that the equation has exactly one solution? Note: This occurs when the expression under the square root sign is zero.

(c) What condition on k will assure that there are two real roots? Note: This occurs when the expression under the square root sign is positive.

20. For a certain predator population, the number P of prey eaten per day depends on the density D of the prey (measured as number per square foot). The relationship is given by

P= 32D / (1 + 3D)

(a) Make a graph of P as a function of D covering values of D up to 5 per square foot.

(b) Explain why it is reasonable that the graph is increasing and concave down.

(c) Find the equation of the horizontal asymptote.

(d) What is the physical significance of the horizontal asymptote?

21. Newton's law of gravity states that the gravitational attraction F between two bodies is proportional to 1 over the square of the distance d between their centers.

(a) Using k as the constant of proportionality, express F as a function of d.

(b) What happens to F near the pole at d = 0?

(c) Explain in practical terms the meaning of the pole at d = 0.

22. The number p of patrons in a restaurant should normally reach its peak over the lunch and dinner hours. Thus it may be appropriate to model p as a quartic function of time. Typical patronage of a certain restaurant is recorded below. Here t is hours since 10 a.m.

t	0	2	3	5	8	10	11
p	5	23	20	14	9	48	34

(a) Make a quartic model for patronage as a function of time since 10 a.m.

(b) You can run your restaurant with fewer staff when the patronage is 10 or less. At what times in the afternoon do you need less staff?

(c) What are the peak business times?