Page 86 - Schaum's Outline of Differential Equations

CHAP. 7] APPLICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS 69

7.29. A mold grows at a rate proportional to its present size. If the original amount doubles in one day, what proportion
of the original amount will be present in five days? Hint: Designate the initial amount by N 0. It is not necessary to
know N 0 explicitly.

7.30. A yeast grows at a rate proportional to its present size. If the original amount doubles in two hours, in how many
hours will it triple?

7.31. The population of a certain country has grown at a rate proportional to the number of people in the country. At
present, the country has 80 million inhabitants. Ten years ago it had 70 million. Assuming that this trend continues,
find (a) an expression for the approximate number of people living in the country at any time t (taking t = 0 to be
the present time) and (b) the approximate number of people who will inhabit the country at the end of the next
ten-year period.

7.32. The population of a certain state is known to grow at a rate proportional to the number of people presently living in
the state. If after 10 years the population has trebled and if after 20 years the population is 150,000, find the number
of people initially living in the state.

7.33. A certain radioactive material is known to decay at a rate proportional to the amount present. If initially there are
100 milligrams of the material present and if after two years it is observed that 5 percent of the original mass has
decayed, find (a) an expression for the mass at any time t and (b) the time necessary for 10 percent of the original
mass to have decayed.

7.34. A certain radioactive material is known to decay at a rate proportional to the amount present. If after one hour it is
observed that 10 percent of the material has decayed, find the half-life of the material. Hint: Designate the initial
mass of the material by N 0. It is not necessary to know N 0 explicitly.

7.35. Find N(t) for the situation described in Problem 7.7.
7.36. A depositor places $10,000 in a certificate of deposit which pays 6 percent interest per annum, compounded
continuously. How much will be in the account at the end of seven years, assuming no additional deposits or
withdrawals?

7.37. How much will be in the account described in the previous problem if the interest rate is 7y percent instead?
7.38. A depositor places $5000 in an account established for a child at birth. Assuming no additional deposits or with-
drawals, how much will the child have upon reaching the age of 21 if the bank pays 5 percent interest per annum
compounded continuously for the entire time period?

7.39. Determine the interest rate required to double an investment in eight years under continuous compounding.

7.40. Determine the interest rate required to triple an investment in ten years under continuous compounding.
7.41. How long will it take a bank deposit to triple in value if interest is compounded continuously at a constant rate of
5^ percent per annum?

7.42. How long will it take a bank deposit to double in value if interest is compounded continuously at a constant rate of
81 percent per annum?

7.43. A depositor currently has $6000 and plans to invest it in an account that accrues interest continuously. What interest
rate must the bank pay if the depositor needs to have $10,000 in four years?

7.44. A depositor currently has $8000 and plans to invest it in an account that accrues interest continuously at the rate of
6 j percent. How long will it take for the account to grow to $13,500?

7.45. A body at a temperature of 0° F is placed in a room whose temperature is kept at 100° F If after 10 minutes
the temperature of the body is 25° F, find (a) the time required for the body to reach a temperature of 50° F, and
(b) the temperature of the body after 20 minutes.