Activity
1. The thickness in millimeters of a folded sheet of paper after it is folded times is given by the equation .
a. What does the number 0.05 represent in the equation?
Compare your answer:
The sheet of paper is 0.05 mm thick.
b. What does the number 2 represent in the equation?
Compare your answer:
The number 2 indicates that the thickness doubles with each increment of , the number of folds.
c. Use graphing technology to graph the equation .
Compare your answer:
Here is a table of values for the thickness of the paper after different numbers of folds:
0 | 0.05 |
1 | 0.1 |
2 | 0.2 |
3 | 0.4 |
4 | 0.8 |
5 | 1.6 |
d. How many folds does it take before the folded sheet of paper is more than 1 mm thick? How many folds before it is more than 1 cm thick? Explain how you know.
Compare your answer:
It takes 5 folds before the paper is more than 1 mm thick. After 4 folds, the paper is 0.8 mm thick, and after 5 folds, it is 1.6 mm thick. It takes 8 folds before the paper is more than 1 cm or 10 mm thick (1.28 cm to be precise).
2. The area of a sheet of paper is 93.5 square inches.
a. Find the remaining visible area of the sheet of paper after it is folded in half once, twice, and three times.
Compare your answer:
46.75, 23.375, 11.6875 square inches
b. Write an equation expressing the visible area of the sheet of paper in terms of the number of times it has been folded, .
Compare your answer:
c. Use graphing technology to graph the equation.
Compare your answer:
d. In this context, can take negative values? Be prepared to show your reasoning.
Compare your answer:
The variable cannot take negative values (unless the original sheet of paper can be unfolded).
e. Can take negative values? Be prepared to show your reasoning.
Compare your answer:
No. The area of the sheet of paper can never be negative, no matter how many times it is folded.
f. Identify the domain of the graph using an inequality.
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g. Identify the range of the graph using an inequality.
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. Note that the area cannot be negative, or even 0, because the paper will not disappear.
h. What equation represents the horizontal asymptote?
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3. Use the equation to make a prediction of the thickness with 7 folds.
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4. The -intercept represents the initial thickness of the paper before folding (when is zero). Calculate the value of the -intercept.
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5. The asymptote represents the possible limit of thickness of the paper as it is folded. What does this mean?
Compare your answer:
In this equation, the value of thickness of the paper becomes larger as the number of folds increases (approaches infinity). This means that, as the paper is folded an increasing number of times, the thickness has the potential to be infinite, but in reality, it can never quite reach it.
Self Check
Additional Resources
Domain and Range of Exponential Graphs
The graph below describes the amount of caffeine, , in a person’s body hours after an initial measurement of 100 mg. The equation of this line is .
What is the smallest value can reach?
Since is the amount of caffeine, it can get close to 0.
Can ever take on negative values?
No, time will not be negative in this relationship because time is moving forward.
What are the lowest and highest values for ?
At the beginning, , and can go as long as it takes for to get so close to 0 it is insignificant.
The domain (or values of ) written as an inequality is .
The range (or values of ) written as an inequality is .
Try it
Try It: Domain and Range of Exponential Graphs
The dollar value of a car is a function, , of the number of years, , since the car was purchased. The car was purchased at $12,000. Tell the domain and range. The equation of this line is .
(is exponent)
Compare your answer:
Here is how to determine the domain and range of an exponential function:
To determine the domain, look at the values. The car is bought when , so the domain is .
To determine the range, look at the values, or the possible dollar values of the car. The car is bought at $12,000 and will lose value until the value is close to $0, but it will never be worth negative dollars.
The range is .