4.1E: Growth and Decay (Exercises)

In Exercises 1-6, a radioactive substance decays at a rate proportional to the amount present at any time t. 

1. The half-life of a radioactive substance is 3200 years. Find the quantity \(Q(t)\) of the substance left at time \(t > 0\) if \(Q(0)=20\) g.

2. The half-life of a radioactive substance is 2 days. Find the time required for a given amount of the material to decay to 1/10 of its original mass.

3. A radioactive material loses 25% of its mass in 10 minutes. What is its half-life?

4. A tree contains a known percentage \(p_0\) of a radioactive substance with half-life \(\tau\). When the tree dies the substance decays and isn’t replaced. If the percentage of the substance in the fossilized remains of such a tree is found to be \(p_1\), how long has the tree been dead?

5. If \(t_p\) and \(t_q\) are the times required for a radioactive material to decay to \(1/p\) and \(1/q\) times its original mass (respectively), how are \(t_p\) and \(t_q\) related?

6. Find the decay constant \(k\) for a radioactive substance, given that the mass of the substance is \(Q_1\) at time \(t_1\) and \(Q_2\) at time \(t_2\).

7. A process creates a radioactive substance at the rate of 2 g/hr and the substance decays at a rate proportional to its mass, with constant of proportionality \(k=.1 (\mbox{hr})^{-1}\). If \(Q(t)\) is the mass of the substance at time \(t\), find \(\lim_{t\to\infty}Q(t)\).

8. A bank pays interest continuously at the rate of 6%. How long does it take for a deposit of \(Q_0\) to grow in value to \(2Q_0\)?

9. At what rate of interest, compounded continuously, will a bank deposit double in value in 8 years?

10. A savings account pays 5% per annum interest compounded continuously. The initial deposit is \(Q_0\) dollars. Assume that there are no subsequent withdrawals or deposits.

1. How long will it take for the value of the account to triple?
2. What is \(Q_0\) if the value of the account after 10 years is $100,000 dollars?

11. A candymaker makes 500 pounds of candy per week, while his large family eats the candy at a rate equal to \(Q(t)/10\) pounds per week, where \(Q(t)\) is the amount of candy present at time \(t\).

1. Find \(Q(t)\) for \(t > 0\) if the candymaker has 250 pounds of candy at \(t=0\).
2. Find \(\lim_{t\to\infty} Q(t)\).

12. Suppose a substance decays at a yearly rate equal to half the square of the mass of the substance present. If we start with 50 g of the substance, how long will it be until only 25 g remain?

13. A super bread dough increases in volume at a rate proportional to the volume \(V\) present. If \(V\) increases by a factor of 10 in 2 hours and \(V(0)=V_0\), find \(V\) at any time \(t\). How long will it take for \(V\) to increase to \(100 V_0\)?

14. A radioactive substance decays at a rate proportional to the amount present, and half the original quantity \(Q_0\) is left after 1500 years. In how many years would the original amount be reduced to \(3Q_0/4\)? How much will be left after 2000 years?

15. Phuong can magically create gold continuously at the rate of 1 ounce per hour, but her assistant JINK steals it continuously at the rate of 5% of however much is there per hour. Let \(W(t)\) be the number of ounces that Phuong has at time \(t\). Find \(W(t)\) and \(\lim_{t\to\infty}W(t)\) if \(W(0)=1\).

16. A process creates a radioactive substance at the rate of 1 g/hr, and the substance decays at an hourly rate equal to 1/10 of the mass present (expressed in grams). Assuming that there are initially 20 g, find the mass \(S(t)\) of the substance present at time \(t\), and find \(\lim_{t\to\infty} S(t)\).

17. A tank is empty at \(t=0\). Water is added to the tank at the rate of 10 gal/min, but it leaks out at a rate (in gallons per minute) equal to the number of gallons in the tank. What is the smallest capacity the tank can have if this process is to continue forever?

18. A person deposits $25,000 in a bank that pays 5% per year interest, compounded continuously. The person continuously withdraws from the account at the rate of $750 per year. Find \(V(t)\), the value of the account at time \(t\) after the initial deposit.

19. A person has a fortune that grows at rate proportional to the square root of its worth. Find the worth \(W\) of the fortune as a function of \(t\) (in years) if it was $1 million six months ago and is $4 million today.

20. Let \(p=p(t)\) be the quantity of a product present at time \(t\). The product is manufactured continuously at a rate proportional to \(p\), with proportionality constant 1/2, and it is consumed continuously at a rate proportional to \(p^2\), with proportionality constant 1/8. Find \(p(t)\) if \(p(0)=100\).

21. A person infected with a virus goes to an isolated island inhabited by 1000 (including himself) people. If the rate of virus spread is proportional to the product of the number of infected people and the number of people not infected, how many people were infected after 6 days if the number of people infected after 4 days is 50.

22. A lake with an estimated limiting population of 6000 fish was stocked with 1000 fish in the year 2000. In 2007 it was estimated that there were 3000 fish. How many fish were in the lake in 2021? 

23. In 2010 there were approximately 700 monkeys living on an island. In 2020 there were about 1000 of them. If the estimated limiting population is 7000, how many monkeys should we expect to be living on the island in 2030?

24. In 1994 there were approximately 500 sea otters living near an island in the pacific. In 2005 there were about 700 otters. If it is estimated that the waters around the island can support a maximum of 3000 otters, find how many otters will be living there in 2025.

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