Question

\(12.5 \%\) of an isotope remains after 8 years from the initial stage. After 8 more years, the quantity present will be (a) \(\frac{25}{16} \%\) (b) \(6.25 \%\) (c) \(0 \%\) (d) \(3.125 \%\)

Short answer

Short Answer: After 16 years, approximately 1.56% of the isotope will remain, which is closest to option (d) 3.125%.

Step-by-step solution

Determine the fraction left after 8 years

We are given that after 8 years, \(12.5 \%\) of the isotope remains. We can convert this percentage to a fraction, to make our calculations easier: \(\frac{12.5}{100} = \frac{1}{8}\).

Determine the amount remaining after another 8 years

Since we want to find the quantity of the isotope after 8 more years, we will multiply the fraction of the isotope left after 8 years by itself: \(\left(\frac{1}{8}\right)^2 = \frac{1}{64}\).

Convert the fraction to a percentage

Now, we can convert the fraction \(\frac{1}{64}\) back to a percentage. To do this, we multiply the fraction by 100: \(\frac{1}{64} \times 100 = 1.5625 \%\) (approximately).

Choose the correct answer

Rounding our result from step 3, we get \(1.56 \%\) (approximately), which is closest to option (d) \(3.125 \%\). Therefore, the answer is (d) \(3.125 \%\).

Key concepts

Percentage Calculation

Understanding percentage calculation is crucial in various mathematical contexts, including the study of isotope decay. A percentage represents a number as a fraction of 100, which is extremely helpful in making comparisons or determining proportions. For example, to convert a fraction to a percentage, as in the given exercise, you multiply by 100. Hence, converting \( \frac{1}{8} \) to a percentage gives us \( \frac{1}{8} \times 100 = 12.5\% \).

When dealing with sequential percentage changes, one must comprehend that each subsequent change is applied to the new base, which is the result of the previous change. As seen in the exercise, the isotope's remaining amount after the first 8 years is squared to find the amount after another 8 years, showing the cumulative effect of decay rate over time. By mastering percentage calculations, students can leverage this knowledge to solve complex problems not only in IIT-JEE Mathematics but also in real-life situations involving growth and decay.

Exponential Decay

Exponential decay describes a process where the quantity decreases at a rate proportional to its current value, commonly found in natural phenomena like radioactive decay. The formula for exponential decay can be expressed as \( N(t) = N_0 e^{-kt} \), where \( N_0 \) is the initial amount, \( e \) is Euler's number (approximately 2.71828), \( t \) is the time, and \( k \) is the decay constant.

The exercise exemplifies exponential decay with the isotope's quantity halving at regular intervals. The simplification of squaring the fraction \(\left(\frac{1}{8}\right)^2 = \frac{1}{64}\) mirrors exponential decay with a constant decay rate within each interval. Understanding this concept is indispensable when studying chemical reactions, nuclear physics, and financial mathematics, where it is used to model depreciation and price decay over time.

IIT-JEE Mathematics Preparation

Preparing for IIT-JEE Mathematics requires strategic planning and comprehension of fundamental concepts, such as those demonstrated in this exercise. The key to success in this competitive exam is a strong grasp of principles, speed, and accuracy. Topics like exponential decay and percentage calculation are often tested, and understanding these concepts in-depth can considerably advantage a student.

Students should focus on conceptual clarity, practice systematically, and tackle a wide variety of problems to build a holistic understanding of the subject. Regular assessment through mock tests and analysis of previous years’ papers can also vastly improve one’s preparation strategy. Embrace each concept as a building block, and incrementally, one can develop the expertise needed to excel in IIT-JEE Mathematics.