Exponential Equations

Algebra 1

Exponential Equations

• A way of demonstrating percent growth or decay in a scenario.

• Interest• Radioactive Decay• Diminishing Value

• Uses the base equation:• y = abx

Growth and Decay

• Present Growthy = A (1 + r ) x

• Present Decayy = A (1 - r ) x

• A new company as of 2015, “Show me the Money”, has an initial worth of 101 thousand dollars. The projected worth of the company can be modeled by using an exponential equation. Through calculations, “Show me the Money” expects to grow by 5% every year. Let x represent the number of years the company has been in business and y represent the total worth of the company. What is the projected worth of the company in 5 years?

y = projected worth x = number of years

y = 101 (1.05) 5

128.9 thousand dollars

• A new company as of 2015, “Show me the Money”, has an initial worth of 101 thousand dollars. The projected worth of the company can be modeled by using an exponential equation. Through calculations, “Show me the Money” expects to grow by 5% every year. Let x represent the number of years the company has been in business and y represent the total worth of the company. When will the company be worth more than 200 thousand dollars? Round your answer the to nearest year.

y = 101 (1.05) 15

15 years

• One airline company, Slingshot, has made an equation to represent their companies worth, in millions of dollars, after every month. The equation is as followed:

y = 4.67(1.0025)x

• How can this equation help you better understand the company. Be specific.

• What will the company be worth in 6 years?

y = 4.67 (1.0025) 72

5.59 million dollars

y = projected worth x = number of years

y = 83 (.86) 5

39.05 thousand dollars

• A new company as of 2015, “Happy Cushions”, has an initial worth of 83 thousand dollars. The projected worth of the company can be modeled by using an exponential equation. Through calculations, “Happy Cushions” expects to depreciate by 14% every year. Let x represent the number of years the company has been in business and y represent the total worth of the company. What is the projected worth of the company in 12 years?

• A new company as of 2015, “Happy Cushions”, has an initial worth of 83 thousand dollars. The projected worth of the company can be modeled by using an exponential equation. Through calculations, “Happy Cushions” expects to depreciate by 14% every year. Let x represent the number of years the company has been in business and y represent the total worth of the company. When will the company be worth less than 1 thousand dollars? Round your answer the to nearest year.

y = 83 (.86) 30

30 years

• A represents the amount that remains of a radioactive material in grams after t years. A can be found using the equation:

A = 50(.67) t

How much radioactive material is left after 3 years?

A = Radioactive Remains t = number of years

A = 50 (.67) 3

15.04 grams

Example

• The price of a can of tuna cost $2.50 at Al’s Grocery Store. If the price of a can of tuna is expected to increase by .75% every year, how much will a can of tuna cost at Mel’s in 2020? (Assume that 2011 is year 0)

Example

• Mr. Jones bought a brand new car for $34,100. Mr. Jones knows the car will decrease in value by 3.5% every year. How much will Mr. Jones car be worth in 5 years?

Back to the Future!!!

• Past Growthy = A( 1 + r ) - x

• Past Decayy = A( 1 - r ) - x

Example

• Al loves history and wants to plan a throw-back week for the store. He plans on choosing different items and using their price from 1925 all week, so the customers can see what it was like in the “old days”. What will be the price of a can of tuna if Al chooses it as one of the items. (Can of Tuna is $2.50 and percent increase is 2%)

Example

• Mrs. Jones has a little bit of a wild-side and decided to buy a used motorcycle instead of a car like her husband. She had to pay $5,400 for the used bike but wants to figure out what the price of the bike was new. If the bike is ten years old and has decreased by 3.5% every year, how much was the bike brand new?