# Solving Exponential Equations, Word Problems (Grades 11-12)

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## Solving Exponential Equations, Word Problems

1.

A certain type of bacteria doubles every hour. If there are 12 bacteria cells to start with, how much time has elapsed if there are now approximately 362 cells?

- 3.4 hours
- 8.5 hours
- 4.9 hours
- 0.7 hours

2.

Mary invested $1,000 in an account which has continuously compounded interest, with a rate of 1.7%. About how long has she left her money in this account, if she currently has $1,202? The formula for continuously compounded interest is [math]A=Pe^{rt}[/math], where [math]P[/math] is the initial amount invested, [math]r[/math] is the interest rate, and [math]t[/math] is the amount of time, in years, the money has been invested.

- 10.8 years
- 1.1 years
- 0.4 years
- 0.2 years

3.

A meteoroid traveling in space just hit Earth's atmosphere. Every 2 seconds, it becomes a tenth smaller. For about how long has the meteoroid been in the Earth's atmosphere, if it began with a mass of 50 kg, and now only has a mass of about 1 kg?

- There is not enough information to determine this.
- 1.7 seconds
- 3.4 seconds
- 7.8 seconds

4.

There is a 100 gram sample of a radioactive substance. After 1 hour, only 63 grams of the sample is left. What is the approximate value of the decay constant, [math]k[/math], if the equation used to determine the decay of this substance is [math]N = N_0 e^{kt} ?[/math]

- -0.46
- 0.46
- 0.04
- 0.07

5.

Tommy is hiking up a mountain. As he hikes higher, the atmosphere becomes thinner and there is less pressure. The atmospheric pressure at sea level is about 101.3 kPa, and the equation [math]P(h) = P_0 e^(-0.00012 h)[/math] gives the atmospheric pressure, in kilopascals, depending on how high above sea level one is, measured in meters. About how high above sea level is Tommy, if the atmospheric pressure is nearly 68 kPa? Round to the nearest meter.

- 347 m
- 3321 m
- 566 m
- 7614 m

6.

Moore's Law states that the number of transistors on a computer chip will double about every 2 years. In the year 2000, there were about 10,000,000 transistors on a computer chip. In approximately what year were there around 1,000,000,000 transistors on a chip? (Assume that the number of transistors on a computer chip doubles exactly every 2 years.)

- 2013
- 2009
- 2007
- 2003

7.

A loaf of bread has started to go moldy. Currently, the moldy spot is very small, only about 0.4% of the total loaf. If the affected portion of the loaf doubles every 8 hours, about how much time will it take for the moldy portion to grow to approximately 5% of the loaf? Round to one decimal.

- 2.6 hours
- 3.6 hours
- 20.2 hours
- 29.2 hours

8.

Brandon Valley University is concerned about their data storage capacity. Their IT department has forecast that the amount of data storage needed will increase by a factor of 10 every 3 years. They currently have about 1,500 TB of data storage capacity, and it is just enough for their current needs. If they increase their current total storage capacity to 5,000 TB, for about how long will this larger storage capacity meet the university's needs? Round to one decimal place.

- 0.5 years
- 1.6 years
- 3.6 years
- 11.1 years

9.

Forest fires can spread very quickly, and their growth can be considered exponential. Although there are many other factors, the growth of a forest fire from a small area can be approximately modeled by the exponential function [math]F(t) = ae^{bt}[/math], where [math]t[/math] is in hours and [math]F[/math] measures the land area affected in square kilometers. If a certain forest fire covered 0.5 square kilometers at 1:00 pm, and then grew to 3.2 square kilometers at 3:00 pm, at about what time would the fire cover 10 square kilometers?

- 4:12 pm
- 6:12 pm
- 8:00 pm
- 9:12 pm

10.

Water hyacinth is a plant that lives on top of water. In many areas, it is invasive. In these areas, without animals that eat it and with abundant food sources, it grows exponentially and can be modeled by the function [math]H(t) = H_0 e^{a \ t}[/math], where [math]H[/math] is the surface area, in square feet, that it covers, and [math]t[/math] is the time, measured in days, that it has been growing. If a certain river has a 4 foot square patch of water hyacinth on it, and then 13 days later it has grown to about 6.5 square feet, how many MORE days will it take for the water hyacinth to cover 12 square feet of the river's surface? Round answers to one decimal place.

- 15.9 days
- 16.7 days
- 27.1 days
- 42.7 days

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