# Interpreting Parameters of Exponential Functions (Grades 11-12)

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## Interpreting Parameters of Exponential Functions

1.

Johanna is working in a lab that deals with radioactive substances. A 100 g sample was left unmarked and she now has to try and determine what it is. She has some guesses, and believes that it will decay according to the function [math]S(t) = 100e^(-0.053t)[/math], where [math]S[/math] is the amount of the sample left, in grams, after [math]t[/math] hours. After some more testing, she believes that it is a different substance that will decay more rapidly. Which of the following functions would be possible alternatives, given this information?

- [math]S(t) = 95 e^(-0.053t)[/math]
- [math]S(t) = 100 e^(-0.041t)[/math]
- [math]S(t) = 100 e^(-0.087t)[/math]
- [math]S(t) = 100e^(0.053t)[/math]

2.

A scientist is studying local rabbits. She has found a sample group, and can predict their short term population growth by the function [math]R(t) = 15*3^t[/math], where [math]R[/math] is the rabbit population after [math]t[/math] years. How many rabbits are there to start with in the sample group?

- 15
- 3
- 45
- It depends on the value of [math]t[/math].

3.

Sally invests some money in a savings account that has compound interest, compounded yearly. If she can track how much money she will have after [math]t[/math] years by the function [math]A(t) = 3000(1.021)^t[/math], what is the interest rate?

- 102.1%
- 30%
- 2.1%
- 1.021 %

4.

The temperature of a hot object cooling to room temperature can be modeled by the function [math]T(t) = a + be^(-kt)[/math], where [math]T[/math] measures the temperature in °C, [math]t[/math] measures the time in hours, and [math]k >= 0[/math].

A.

The temperature of the object will eventually cool to room temperature, given enough time. Which parameter(s) would therefore most likely represent room temperature?

- [math]a[/math]
- [math]b[/math]
- [math]k[/math]
- [math]a/b[/math]

B.

Each object will have a different rate at which it cools. Depending on the type of material, its geometry, and other factors, it will cool faster or slower. Thus, two different objects, like a cup of coffee and a loaf of banana bread, even if they start at the same temperature and are in the same room, will cool at different rates. Which parameter(s) would most likely represent this cooling rate?

- [math]a[/math]
- [math]b[/math]
- [math]k[/math]
- [math]-kt[/math]

C.

The difference of the object's temperature and the surrounding temperature will impact how quickly the object cools. If the difference is greater (meaning the object is hotter), it will take longer. Which parameter(s) would most likely represent this difference?

- [math]a[/math]
- [math]b[/math]
- [math]k[/math]
- [math]t[/math]

5.

Carbon dating uses the fact that carbon-14, an element found in nearly all living things, decays exponentially. The function [math]C(t) = 5 e^(-0.00012t)[/math], models the approximate amount of carbon-14, in micrograms, left in a certain sample of a plant after it died. What does the value 5 represent in this model?

- The decay rate constant of carbon-14.
- The half life of carbon-14.
- The amount of carbon-14 originally in the specimen, in micrograms.
- The amount of carbon-14 left in the specimen, in micrograms, after [math]t[/math] years.

6.

An important part of a nuclear fission reactor is the chain reaction process. In simplified terms, this process starts with a single neutron which is shot into fissile material, such as uranium. This neutron eventually hits and is absorbed by a uranium atom. The uranium atom is unstable with the extra neutron and eventually splits apart. This splitting releases additional neutrons, as well as energy, and the process repeats with each of these new neutrons. If each time a neutron hits an atom, two neutrons are released, which of the following functions correctly models this simplified example?

- [math]N(t) = 2^t[/math], where [math]N[/math] is the number of loose neutrons and [math]t[/math] is the time, in seconds.
- [math]N(a) = 2^a[/math], where [math]N[/math] is the number of loose neutrons, and [math]a[/math] is the number of uranium atoms that have been hit.
- [math]N(a) = 2*2^a[/math], where [math]N[/math] is the number of loose neutrons, and [math]a[/math] is the number of uranium atoms that have been hit.
- [math]N(t) = e^(2t)[/math], where [math]N[/math] is the number of loose neutrons and [math]t[/math] is the time, in seconds.

7.

Moore's law states that the number of transistors on a computer chip will double about every two years. The function [math]N(t) = (2.5xx10^6) * 2^(t \ //18)[/math] predicts the approximate number of transistors in a computer chip [math]t[/math] months after the year 1990. Exactly how long does this function predict it will take for the number of transistors on a computer chip to double?

- Every 24 months.
- Every 18 months.
- Every 2 months.
- It can't be determined without knowing the number of transistors on a computer chip at some point beyond the year 1990.

8.

Often, things shared over social media can be spread exponentially. The function [math]N(t) = 5*3^t[/math] models the approximate number of people, [math]N[/math], who have read and then shared a certain news article over a period of [math]t[/math] hours. According to this function, with about how many people will each person share a news article, on average, every hour?

- Not enough information to determine.
- 5
- 3
- 1

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