Modeling Exponential Growth Using Two Points
MTH 111c
Modeling Exponential Growth Using Two Points
EXERCISES
1. For each part of this problem, find the equation of an exponential function that passes through the given pair of points, then verify your answer using your graphing calculator.
a. (1, 18) and (3, 1458)
b. (-1, 3/4) and (2, 48)
c. (2, 7) and (5, 3)
d. (0, 3) and (5, 100)
e. (3, 8) and (7, 24)
2. The graph of a function passes through the points (0, 3) and (2, 12). Is the function linear or exponential?
3. Let h(0) =2 and h(1) = 6. Find a formula for h, assuming h is:
a. linear. b. exponential.
4. In 1965, there were 60,000 rabbits in a small country. By 1967, the number had increased to 2,400,000. Assume that the number of rabbits increased exponentially with the number of years that have elapsed since 1965.
a. Write an equation for this function. Don’t forget to define your variables.
b. How many rabbits would you predict in 1970?
c. According to your model, when was the first pair of rabbits introduced into this country?
5. A rule of thumb used by car dealers is that the trade-in value of a car decreases by 30% each year. That is, the value at the end of any year is 70% of its value at the beginning of that year.
a. Suppose that you own a car whose trade-in value is presently $2350. How much will it be worth 1 year from now? 2 years from now? 3 years from now?
b. Write an equation for this function.
c. In how many years from now should the trade in value be $600?
d. If the car is presently 2.7 years old, what was its trade-in value when it was new?
e. The car cost $7430 when it was new. How do you explain the discrepancy between this number and your answer to part d?
6. Elijah is driving along a straight, level highway at 64 kilometers per hour (km/h) when his car runs out of gas. As he slows down, his speed decrease exponentially with the number of seconds since he ran out of gas, dropping to 48 km/h after 10 seconds.
a. Write an equation for this function.
b. Predict Elijah’s speed after 25 seconds.
c. At what time will Elijah’s speed be 10 km/h?
d. Draw the graph of the function for speed in the domain from 0 through the time when Elijah reaches 10 km/h.
e. What would the actual speed-time graph look like for negative values of time?
f. Explain why this mathematical model would not give reasonable answers at very large values of time.
7. In a fog, the tail lights of the car in front of you seem to appear suddenly. According to Beer’s Law of Radiation Absorption, the intensity of the light varies exponentially with the distance between you and the other car. Suppose that the intensity is 128 units when the car is “right at” your car (i.e. , the distance is essentially zero). At 43 feet, the intensity is only one half of that value.
a. What is the intensity at 86 feet?
b. Write an equation for this function.
c. If the lights are just visible when the intensity is 3 units, how far away should the car be?
d. By what factor does the intensity increase when the car nears from 86 feet away to 43 feet away? From 1043 feet away to 1000 feet away?
e. Why do the lights seem to appear so suddenly?
8. During the first stages of an epidemic, the number of sick people increase exponentially with time. Suppose that at time t = 0 days, there are 40 people sick. By the time t = 3, 200 people are sick.
a. How many people will be sick by the time t = 6?
b. Let s(t) be the number of sick people at time t (in days). Find an equation expressing s(t) in terms of t.
c. Predict the number of sick people at the end of the first week.
d. At what time t does the number of sick people reach 7000?
SOLUTIONS
1 a. [pic]
b. [pic]
c. [pic]
d. [pic]
e. [pic]
2. The function could be either linear or exponential. Given only two points on a curve, you cannot tell what type of curve it is.
3 a. [pic]
b. [pic]
4 a. [pic] where R is the number of rabbits t years since 1965.
b. I would predict about 607 million rabbits in 1970.
c. According to my model, the first pair of rabbits was introduced in 1959. (This can be solved graphically. Remember to show the graph used to answer this question.)
5 a. One year from now, the car will be worth $1645. Two years from now, it will be worth $1151.50. Three years from now, it will be worth $806.05.
b. The equation for this function is [pic] where V(t) is the value of the car t years from now.
c. The trade in value should be $600 in about 3.8 years. (This can be solved graphically. Remember to show the graph used to answer this question.)
d. If the car is presently 2.7 years old, its trade-in value when it was new was about $6156.
e. If the car cost $7430 when it was new, but its trade-in value was only about $6156, this is probably due to the car dealer’s profit.
6 a. An approximate equation for this function is [pic] where s(t) is Elijah’s speed at time t seconds after Elijah’s car ran out of gas.
b. Elijah’s speed after 25 seconds is about 31 km/h.
c. Elijah’s speed will be 10 km/h about 65 seconds after his car runs out of gas.
d.
[pic]
e. For negative values of time, the actual speed graph would be a horizontal line at [pic] because before he ran out of gas, Elijah was driving at a constant speed of 64 km/h.
f. This model would not give reasonable answers at very large values of time because according to the model, he never completely stops.
7 a. At 86 feet, the intensity is 32 units.
b. An approximate equation of this function is [pic] where I is the intensity of the light when the car is d feet away from your car.
c. The car would be about 233 ft away when the intensity is 3 units. (This can be solved graphically).
d. When the car nears from 86 feet away to 43 feet away, the intensity increase by a factor of 2. When the car nears from 1043 feet away to 1000 feet away, the intensity increase by a factor of 2, also.
e. At 30 mph (45 ft/s), the intensity doubles each second, which is very rapid. Thus, the tail lights of the car in front of you seem to appear suddenly.
8 a. By the time t = 6, there will be 1000 sick people
b. [pic]
c. At the end of the first week, there will be about 1710 sick people.
d. The number of sick people reaches 7000 after about 9.63 days. (This can be solved graphically).
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[pic]
[pic]
Speed (km/hr)
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