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Wize High School Algebra II Textbook (Common Core) > Exponential Functions

Applications of Exponential Functions (Growth & Decay)

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Applications of Exponential Functions

Compound Interest

A=P(1+in)nt\boxed{A=P\Bigg(1+\dfrac{i}{n}\Bigg)^{nt}} A=Final Amount ($)P=Initial Amount ($)i=Interest raten=Compound periodst=Time elapsed\begin{array}{rcl} A&=&\text{Final Amount (\$)}\\\\ P&=&\text{Initial Amount (\$)}\\\\ i&=&\text{Interest rate}\\\\ n&=&\text{Compound periods}\\\\ t&=&\text{Time elapsed} \end{array}

Exponential Growth, k>1\colorOne{k>1}

P(t)=a(b)kt\boxed{P(t)=a(b)^{kt}} P(t)=Final Valuea=Initial Value ($)b=Growth Ratek=Growth Constantt=Time elapsed\begin{array}{rcl} P(t)&=&\text{Final Value}\\\\ a&=&\text{Initial Value (\$)}\\\\ b&=&\text{Growth Rate}\\\\ k&=&\text{Growth Constant}\\\\ t&=&\text{Time elapsed} \end{array}


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Exponential Decay, k<1\colorOne{k<1}

P(t)=a(b)kt\boxed{P(t)=a(b)^{kt}} P(t)=Final Valuea=Initial Value ($)b=1−Decay Ratek=Decay Constantt=Time elapsed\begin{array}{rcl} P(t)&=&\text{Final Value}\\\\ a&=&\text{Initial Value (\$)}\\\\ b&=&1-\text{Decay Rate}\\\\ k&=&\text{Decay Constant}\\\\ t&=&\text{Time elapsed} \end{array}

Richter Scale

I=10R\boxed{I=10^R} I=IntensityR=Richter scale magnitude\begin{array}{rcl} I&=&\text{Intensity}\\\\ R&=&\text{Richter scale magnitude} \end{array}

ph Scale

H+=−10pH\boxed{\text{H}^+=-10^{\text{pH}}} H+=Hydrogen Ion ConcentrationpH=pH unit\begin{array}{rcl} \text{H}^+&=&\text{Hydrogen Ion Concentration}\\\\ \text{pH}&=&\text{pH unit} \end{array}

Decibel Scale

I=10dB\boxed{I=10^{dB}} I=IntensitydB=Decibel unit\begin{array}{rcl} I&=&\text{Intensity}\\\\ dB&=&\text{Decibel unit} \end{array}

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Example: Applications of Exponential Functions

Anita invested $4000 into an account that offers 6.25% compounded monthly. How much is in the account after 10 years?
Round all answers to the nearest hundredth.

Let t=10t=10 years.

A(t)=P(1+in)nt=4000(1+0.062512)12(10)=$7460.87\begin{array}{rcl} A(t)&=&P\Big(1+\frac{i}{n}\Big)^{nt}\\\\ &=&4000\Big(1+\frac{0.0625}{12}\Big)^{12(10)}\\\\ &=&\$7460.87 \end{array}

Practice: Applications of Exponential Functions

The half-life of a certain radioactive substance is modelled by the function A(t)=10.4(0.5)t6A(t)=10.4(0.5)^{\frac{t}{6}}, where there is initially 10.4 grams of the substance, the half-life is 6 weeks, and A(t)A(t) is the amount of the substance after tt weeks.

How much of the substance is there after 4 months? Use the fact that there is 4.354.35 weeks in a month.

Practice: Applications of Exponential Functions

The size of a certain species increases five-fold every 3 days. If the size of the population grew to 150000 after 15 days, what was the initial population size?

Practice: Applications of Exponential Functions

The value of a car depreciates roughly 15% annually. Joel buys a new 2021 Honda Civic for $35000 and plans on selling his car again in 10 years. How much will the Honda be worth in 10 years?
Extra Practice
Applications of Exponential Functions
A population of insects increases 6-fold every 3 days. How many species will there be in 10 days if the initial population was 500?
Applications of Exponential Functions
Practice: Applications of Exponential Functions
Marcel has $10000 saved and he wants to invest his money into a low-risk mutual fund. He has two options to choose from:
Invest $10000 at 1.4% compounded bi-weekly or;
Applications of Exponential Functions
Practice: Applications of Exponential Functions
The value of an antique appreciates roughly 3.2% annually. Maggie buys her antique for $5400 and plans on selling it again in 20 years. How much will Maggie profit off her antique?
Applications of Exponential Functions
Practice: Applications of Exponential Functions
The size of a certain species increases four-fold every 17 days. If the size of the population grew to 35 after 8 days, what was the initial population size?
Applications of Exponential Functions
Practice: Applications of Exponential Functions
The size of a certain species increases nine-fold every 6 days. If the size of the population grew to 400,500 after 10 days, what was the initial population size?
Applications of Exponential Functions
Practice: Applications of Exponential Functions
The half-life of a certain radioactive substance is modeled by the function A(t)=5.63(0.5)t9.5A(t)=5.63(0.5)^{\frac{t}{9.5}}, where there is initially 5.63 grams of the substance, the half-life is 9.5 months, and A(t)A(t) is the amount of the substance after tt weeks.
How much of the substance is there after 32 weeks? Use the fact that there is 4.34.3 weeks in a month.
Applications of Exponential Functions
Practice: Applications of Exponential Functions
The half-life of a certain radioactive substance is modelled by the function A(t)=4.7(0.5)t3.4A(t)=4.7(0.5)^{\frac{t}{3.4}}, where there is initially 4.7 micrograms of the substance, the half-life is 3.4 weeks, and A(t)A(t) is the amount of the substance after tt weeks.
How much of the substance is there after 2 months? Use the fact that there is 4.34.3 weeks in a month.