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Chapter 9: Solving Application Problems
Objectives:
- Exponential growth/growth models
- Using logarithms to solve
Steps for Solving Application Problems:
- Read, throw out nonsense numbers
- Assign a variable (What is it asking for?)
- Write an equation
- Solve the equation
- Check, does it make sense?
Exponential Growth Models
An exponential equation or exponential function is of the form y = a' orf{x) = cf, where a>0,a^ 1.
Exponential Growth or Decay Formula: _P(t) = P^t/*,a>0,a ^_
Po represents the original amount present, P{t) represents the amount present after t years, and a and k are constants.
When a> 1, P{t) increases. (Growth) Ex: a = 2
When 0 < (3 < 1, P{t) decreases. (Decay) Ex: a = Vi X M 7 6 5
1 ~
i '» 1 4 4 J{2.^ 4)^^1 2 Hi. ) i
0 2 ^ (^) ] 2. 4 5^2 .
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Ex: The exponential graph below models the U.S. cellular telephone subscribership, P(t) , in thousands, for / years 1989 through 2008. The formula P(t) = 3,500(1.257)* models this growth.
a. Use the formula to calculate the number of subscribers in 1989.
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b. Use the formula to calculate the year it will be when the number of subscribers reaches 500, thousand. ,. V-"
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You try:
- The exponential graph below models the percentage of surface sunlight, Av), that reaches a depth of jc feet beneath the surface of the ocean. The formula / ( x ) = 20(0.975)'' models this decay.
a. Use the formula to calculate the percentage of surface sunlight intensity at a depth of 20 feet.
Intensity of Sunlight Beneath the Ocean's Surface
b. Use tlie tomula to calculate the depth needed to only have 1% of surface sunlight intensity.
'\y —^ 40 60 80 100 120 Ocean Depth (feet)
**Note: When the a in the exponential function,/^) = is replaced with a very special letter e, we get the natural
exponential function, fix) = e^. Since e> 1, we use f{x) =. When k > 0, f(x) increases as x increases and we
have exponential growth. When k < 0,f(x) decreases as x increases and we have exponential decay.**
Natural Exponential Growth or Decay Formula: P{t) = P^e'"
Ex: The exponential graph below models the risk of having a car accident, R(x) (as a percentage), with respect to a person's blood alcohol concentration, x. The formula R{x) = 6e'"^^^ models this growth.
a. Use the fonnula to calculate the percent of risk of getting into a car accident for a person that has a blood alcohol concentration around 0.05.
b. Use the formula to calculate the blood alcohol concentration necessary to have a 100% risk of getting into a car accident. njw
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A bl«od ileoliol eoiie«n(r«ti»n of 0. eorretpoRdi to w a r eertainfy, or a 100% probibilitjf, of a ear aeeiiieRt.
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20%
0.05 0.10 0.15 0.20 0. Blood Alcohol Concentration
You try:
- T he exponential graph below shows the U.S. gross domestic product (GDP), the market value of all goods and services produced within the US, P{t) , in billions, for / years between 1965 and 2010. The general formula P(t) = P^e'" models this growth.
a. Use the graph to find the initial GDP,.
h. Substitute a point on the graph into the formula P(t) = Pe'" to find the constant grow rate, k.
c. Use PQ and k to find the general formula P(0 =£jBt
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urn mm I 'Kimi - •
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(^0) i<Ki5 i<)7o i')7s list) m!i I 'm mm Year
d. Use the formula found in part c. to calculate the GDP in 2013.
e. Use the formula found in part c. to find when the GDP will be $20,000 billion.
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Plutonium-239, a radioactive material used in most nuclear reactors, decays exponentially. If there are originally 16 grams of plutonium-239, then the amount of plutonium-239, P(t), remaining after / years is modeled by the formula P{t) = 16e*', where k<Q, since the amount of plutonium decreases as time goes on.
a. If approximately 3.995 grams of plutonium-239 remain after 50,000 years, find the decay rate, k. And, state the fimction that models this case.
b. How much plutonium will remain after 50 years?
c. How long will it take to have only 2 grams remain?
m = 2
Id: 1( Q
