Select the correct answer.
A scientist has a 120 mg sample of an unstable isotope. The amount of the substance remaining in the sample decreases at a rate of $16 \%$ each day. At the end of $t$ days, there is less than 30 mg of the substance remaining.
Which inequality represents this situation, and after how many days will the sample size remaining be less than 30 mg?
A. $120(0.984)^t<30 ; t=86$ days
B. $120(0.84)^t<30 ; t=7$ days
C. $120(0.84)^t<30 ; t=8$ days
D. $120(0.984)^t<30 ; t=87$ days
Answer (2)
Set up the inequality: 120 ( 0.84 ) t < 30 , representing the decay of the isotope.
Simplify the inequality: ( 0.84 ) t < 0.25 by dividing both sides by 120.
Solve for t using logarithms: \frac{ln(0.25)}{ln(0.84)} \approx 7.95"> t > l n ( 0.84 ) l n ( 0.25 ) ≈ 7.95 .
Round up to the nearest whole number: t = 8 days, since we need the number of days when the remaining amount is less than 30 mg. The final answer is 8 .
Explanation
Understanding the Problem We are given an initial amount of 120 mg of an unstable isotope that decays at a rate of 16% each day. We want to find the inequality that represents the situation when the remaining amount is less than 30 mg, and the number of days it takes for this to happen.
Setting up the Inequality The amount of the substance remaining after t days can be modeled by the equation: R e mainin g = I ni t ia l × ( 1 − Dec a y R a t e ) t In this case, the initial amount is 120 mg, and the decay rate is 16% or 0.16. So the equation becomes: R e mainin g = 120 × ( 1 − 0.16 ) t = 120 × ( 0.84 ) t We want to find when the remaining amount is less than 30 mg, so we set up the inequality: 120 ( 0.84 ) t < 30
Solving for t Now we need to solve for t . First, divide both sides of the inequality by 120: ( 0.84 ) t < 120 30 = 0.25 To solve for t , we can take the logarithm of both sides. Using the natural logarithm (ln): l n ( 0.84 ) t < l n ( 0.25 ) t × l n ( 0.84 ) < l n ( 0.25 ) Since l n ( 0.84 ) is negative, when we divide both sides by l n ( 0.84 ) , we need to reverse the inequality sign: \frac{ln(0.25)}{ln(0.84)}"> t > l n ( 0.84 ) l n ( 0.25 )
Finding the Number of Days Using a calculator, we find: \frac{ln(0.25)}{ln(0.84)} \approx 7.95"> t > l n ( 0.84 ) l n ( 0.25 ) ≈ 7.95 Since t represents the number of days, we need to round up to the nearest whole number because we want to find the number of days when the remaining amount is less than 30 mg. Therefore, t = 8 days.
Final Answer The correct inequality is 120 ( 0.84 ) t < 30 , and it will take 8 days for the sample size remaining to be less than 30 mg.
Examples
Radioactive decay is used in various applications, such as carbon dating to determine the age of ancient artifacts. The concept of exponential decay, similar to the one in this problem, helps scientists estimate the age of organic materials by measuring the remaining amount of carbon-14, a radioactive isotope. Understanding exponential decay is also crucial in medicine for determining the dosage and timing of radioactive treatments, ensuring effective therapy while minimizing harm to the patient. This mathematical model allows for precise calculations and informed decisions in these critical fields.
The inequality representing the decay of the isotope is 120 ( 0.84 ) t < 30 , and it will take 8 days for the sample size remaining to be less than 30 mg. The correct option from the choices is C .
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