10. A radioactive substance decays ov \( 14 \% \). How mery year mass decreases bears will it take for 500 kg of the substance to decay to a mass loss than 200kg? 11. Martyn has some money to invest and sees this advert. Bank Double your money in 15 years. The average annual growth for your investment is \( 4.5 \% \) Will Martyn double his money in 15 years by investing his money with "Bank of Maths?" You must show your workings. MAS 2014
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To solve the first question, you can use the exponential decay formula: \( A = A_0 e^{-kt} \). If a substance decays by \( 14 \%\), then you are left with \( 86\% \) of the original mass each year. Setting up the equation to find out how many years it takes to reduce the mass from \( 500 \, kg \) to less than \( 300 \, kg \) leads to the inequality \( 500 \times (0.86)^t < 300 \). Solving this logarithmically gives \( t > 8.1 \) years, so it would take more than 8 years for the mass to decay to less than \( 300 \, kg \). For Martyn's investment, the rule of 72 is a handy tool. To estimate how long it takes to double your money, divide 72 by the annual interest rate. Here, \( 72 / 4.5 \approx 16 \) years. Since this is longer than 15 years, Martyn will not double his money in the promised timeline. Instead, with a \( 4.5\% \) rate, his investment will grow, but not quite double in that time frame! So, it’s vital to check those growth promises carefully.