Exponentials: Which Model?

Which model is it?

Determining when to use each exponential model

Multiple Choice

For the situation below, identify the type of equation you would apply.

Your new computer costs $174 (It’s a Chromebook ☹️) but it loses value by about 11% each year.

• $$y=a\left(b\right)^x$$

  b = 1 + r

  (appreciate)

• $$y=a\left(b\right)^x$$

  b = 1 - r

  (depreciate)

• $$y=a\left(b\right)^x$$

  $$b=\frac{\ 1}{2}$$

  x = (x/ half-life)

  (half-life)

• $$y=a\left(e\right)^{rt}$$

  b = e

  the natural base

  (continuous)

Multiple Choice

For the situation below, identify the type of equation you would apply.

Phosphorus-32 is used to study a plant’s use of fertilizer.     

It has a half-life of  14.3  days and we have a 75 g sample. 

• $$y=a\left(b\right)^x$$

  b = 1 + r

  (appreciate)

• $$y=a\left(b\right)^x$$

  b = 1 - r

  (depreciate)

• $$y=a\left(b\right)^x$$

  $$b=\frac{\ 1}{2}$$

  x = (x/ half-life)

  (half-life)

• $$y=a\left(e\right)^{rt}$$

  b = e

  the natural base

  (continuous)

Multiple Choice

For the situation below, identify the type of equation you would apply.

The population of St. Louis in 1950 was 856,796. Since then, it has been decreasing; in 2010, the population fell to 319,365. Determine the rate of decline in the population.

• $$y=a\left(b\right)^x$$

  b = 1 + r

  (appreciate)

• $$y=a\left(b\right)^x$$

  b = 1 - r

  (depreciate)

• $$y=a\left(b\right)^x$$

  $$b=\frac{\ 1}{2}$$

  x = (x/ half-life)

  (half-life)

• $$y=a\left(e\right)^{rt}$$

  b = e

  the natural base

  (continuous)

Multiple Choice

For the situation below, identify the type of equation you would apply.

Mrs. Lindsey bought a Scandinavian bookshelf and its value is declines at a rate of 10% per year. After 8 years it was worth $13,000. 

• $$y=a\left(b\right)^x$$

  b = 1 + r

  (appreciate)

• $$y=a\left(b\right)^x$$

  b = 1 - r

  (depreciate)

• $$y=a\left(b\right)^x$$

  $$b=\frac{\ 1}{2}$$

  x = (x/ half-life)

  (half-life)

• $$y=a\left(e\right)^{rt}$$

  b = e

  the natural base

  (continuous)

Multiple Choice

For the situation below, identify the type of equation you would apply.

Suppose Mr. Covey purchased land in 2005 for $25,000.

His land is in San Fran and it gains value annually at 4.2%.

• $$y=a\left(b\right)^x$$

  b = 1 + r

  (appreciate)

• $$y=a\left(b\right)^x$$

  b = 1 - r

  (depreciate)

• $$y=a\left(b\right)^x$$

  $$b=\frac{\ 1}{2}$$

  x = (x/ half-life)

  (half-life)

• $$y=a\left(e\right)^{rt}$$

  b = e

  the natural base

  (continuous)

Multiple Choice

For the situation below, identify the type of equation you would apply.

Anthony invested in an account paying 5% interest compounded continuously. If he’d like the account to have $32,000 so he can buy a new Daytona Charger in 5 years, how much should he invest?

• $$y=a\left(b\right)^x$$

  b = 1 + r

  (appreciate)

• $$y=a\left(b\right)^x$$

  b = 1 - r

  (depreciate)

• $$y=a\left(b\right)^x$$

  $$b=\frac{\ 1}{2}$$

  x = (x/ half-life)

  (half-life)

• $$y=a\left(e\right)^{rt}$$

  b = e

  the natural base

  (continuous)

Multiple Choice

For the situation below, identify the type of equation you would apply.

 The world population in 2000 was approximately 6.08 billion.

The annual rate of increase was about 1.26%.

• $$y=a\left(b\right)^x$$

  b = 1 + r

  (appreciate)

• $$y=a\left(b\right)^x$$

  b = 1 - r

  (depreciate)

• $$y=a\left(b\right)^x$$

  $$b=\frac{\ 1}{2}$$

  x = (x/ half-life)

  (half-life)

• $$y=a\left(e\right)^{rt}$$

  b = e

  the natural base

  (continuous)

Multiple Choice

For the situation below, identify the type of equation you would apply.

A painting was purchased at an art exhibition.

Afterwards, it's value increases by 15% each year.

• $$y=a\left(b\right)^x$$

  b = 1 + r

  (appreciate)

• $$y=a\left(b\right)^x$$

  b = 1 - r

  (depreciate)

• $$y=a\left(b\right)^x$$

  $$b=\frac{\ 1}{2}$$

  x = (x/ half-life)

  (half-life)

• $$y=a\left(e\right)^{rt}$$

  b = e

  the natural base

  (continuous)

Poll

At present, which model do you feel is the most challenging to work with?

• $$y=a\left(b\right)^x$$

  b = 1 +/- r

  (appreciate/depreciate)

• $$y=a\left(b\right)^x$$

  $$b=\frac{\ 1}{2}$$

  x = (x/ half-life)

  (half-life)

• $$y=a\left(e\right)^{rt}$$

  b = e

  the natural base

  (continuous)

• None! They all make sense!

• None!

  They all make sense!

Closing Thoughts:

What information helps you determine which variation to use for exponential modeling problems?

Reminder, the models are:

Appreciate/Depreciate: y = a(b)^x where b = 1 +/- r

Natural/Continuous: y = a(e)^rt where the rate is part of the exp.

Half-Life Decay: y=a(b)^x/half-life where b=1/2