A patient ingests 5 grams of a certain drug which dissolves away at a constant rate of 10% per hour. How long will it take before there is only one gram left?
a.) 15.9 hours
b.) 15.3 hours
Answer (2)
Set up the exponential decay equation: A ( t ) = A 0 ( 1 − r ) t .
Substitute the given values: 1 = 5 ( 0.9 ) t .
Solve for t by taking the natural logarithm of both sides: l n ( 0.2 ) = t l n ( 0.9 ) .
Calculate t : t = l n ( 0.9 ) l n ( 0.2 ) ≈ 15.3 hours. The final answer is 15.3 hours .
Explanation
Problem Setup We are given an exponential decay problem where a patient ingests 5 grams of a drug, and it dissolves at a rate of 10% per hour. We want to find out how long it takes until only 1 gram of the drug remains.
Define Variables Let's define the variables: A ( t ) = amount of drug remaining after t hours A 0 = initial amount of drug (5 grams) r = decay rate (10% = 0.1)
Exponential Decay Equation The exponential decay equation is given by: A ( t ) = A 0 ( 1 − r ) t
Substitute Values We are given A 0 = 5 grams, r = 0.1 , and we want to find the time t when A ( t ) = 1 gram. Substituting these values into the equation, we get: 1 = 5 ( 1 − 0.1 ) t
Simplify Simplify the equation: 1 = 5 ( 0.9 ) t
Isolate Exponential Term Divide both sides by 5: 0.2 = ( 0.9 ) t
Take Natural Logarithm Take the natural logarithm of both sides: l n ( 0.2 ) = l n ( 0. 9 t )
Apply Power Rule of Logarithms Use the power rule of logarithms: l n ( 0.2 ) = t l n ( 0.9 )
Solve for t Solve for t :
t = l n ( 0.9 ) l n ( 0.2 ) Using a calculator, we find: t ≈ 15.2755
Final Answer Therefore, it will take approximately 15.3 hours for the drug to decay to 1 gram.
Examples
Exponential decay is a fundamental concept in various fields. For instance, in finance, it models the depreciation of an asset over time. Imagine a car initially valued at $20,000 depreciates at a rate of 15% per year. Using the exponential decay formula, one can predict the car's value after a certain number of years, aiding in financial planning and investment decisions. Similarly, in environmental science, it helps in estimating the decay of pollutants in the environment, crucial for assessing environmental impact and implementing remediation strategies.
The drug will dissolve to 1 gram in approximately 15.3 hours, according to the exponential decay model. The calculations show that starting from 5 grams and decaying at a rate of 10% per hour leads to this result. Thus, the correct answer is option (b) 15.3 hours.
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