Q4.1.1
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.
- How long will it take for the value of the account to triple?
- 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 \(t\).
- Find \(Q(t)\) for \(t > 0\) if the candymaker has 250 pounds of candy at \(t=0\).
- 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. A wizard creates gold continuously at the rate of 1 ounce per hour, but an assistant steals it continuously at the rate of 5% of however much is there per hour. Let \(W(t)\) be the number of ounces that the wizard 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 \(t\) if it was $1 million 6 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. In the situation of Example 4.1.6 find the exact value P(t) of the person’s account after t years, where t is an integer. Assume that each year has exactly 52 weeks, and include the year-end deposit in the computation.
HINT: At time t the initial $1000 has been on deposit for \(t\) years. There have been \(52t\) deposits of $\(50\) each. The first $\(50\) has been on deposit for \(t − 1/52\) years, the second for \(t − 2/52\) years ... in general, the j th $\(50\) has been on deposit for \(t − j/52\) years (\(1 ≤ j ≤ 52t\)). Find the present value of each $\(50\) deposit assuming \(6\)% interest compounded continuously, and use the formula \[1+x+x^{2}+ . . . + x^{n}=\frac{1-x^{n+1}}{1-x}(x\neq 1)\] to find their total value.
b. Let
\[p(t)={Q(t)-P(t)\over P(t)}\]
be the relative error after \(t\) years. Find
\[p(\infty)=\lim_{t\to\infty}p(t).\]
22. A homebuyer borrows \(P_0\) dollars at an annual interest rate \(r\), agreeing to repay the loan with equal monthly payments of \(M\) dollars per month over \(N\) years.
a. Derive a differential equation for the loan principal (amount that the homebuyer owes) \(P(t)\) at time \(t>0\), making the simplifying assumption that the homebuyer repays the loan continuously rather than in discrete steps. (See Example 4.1.6.)
b. Solve the equation derived in (a).
c. Use the result of (b) to determine an approximate value for \(M\) assuming that each year has exactly 12 months of equal length.
d. It can be shown that the exact value of \(M\) is given by
\[M={rP_0\over 12}\left(1-(1+r/12)^{-12N}\right)^{-1}.\]
Compare the value of \(M\) obtained from the answer in (c)
to the exact value if (i) \(P_0=\$50,000\), \(r=7{1\over2
}\)%, \(N=20\) (ii) \(P_0=\$150,000\), \(r=9.0\)%, \(N=30\).
23. Assume that the homebuyer of Exercise 4.1.22 elects to repay the loan continuously at the rate of \(\alpha M\) dollars per month, where \(\alpha\) is a constant greater than 1. (This is called accelerated payment.)
- Determine the time \(T(\alpha)\) when the loan will be paid off and the amount \(S(\alpha)\) that the homebuyer will save.
- Suppose \(P_0=\$50,000\), \(r=8\)%, and \(N=15\). Compute the savings realized by accelerated payments with \(\alpha=1.05,1.10\), and \(1.15\).
24. A benefactor wishes to establish a trust fund to pay a researcher’s salary for \(T\) years. The salary is to start at \(S_0\) dollars per year and increase at a fractional rate of \(a\) per year. Find the amount of money \(P_0\) that the benefactor must deposit in a trust fund paying interest at a rate \(r\) per year. Assume that the researcher’s salary is paid continuously, the interest is compounded continuously, and the salary increases are granted continuously.
25. A radioactive substance with decay constant \(k\) is produced at the rate of
\[{at\over1+btQ(t)}\]
units of mass per unit time, where \(a\) and \(b\) are positive constants and \(Q(t)\) is the mass of the substance present at time \(t\); thus, the rate of production is small at the start and tends to slow when \(Q\) is large.
- Set up a differential equation for \(Q\).
- Choose your own positive values for \(a\), \(b\), \(k\), and \(Q_0=Q(0)\). Use a numerical method to discover what happens to \(Q(t)\) as \(t\to\infty\). (Be precise, expressing your conclusions in terms of \(a\), \(b\), \(k\). However, no proof is required.)
26. Follow the instructions of Exercise 4.1.25, assuming that the substance is produced at the rate of \(at/(1+bt(Q(t))^2)\) units of mass per unit of time.
27. Follow the instructions of Exercise 4.1.25, assuming that the substance is produced at the rate of \(at/(1+bt)\) units of mass per unit of time.
