Solved: A detective finds a murder victim at 9 am. The temperature of the body is

Match the graphs in Figure 11.50 with the following descriptions. (a) The temperature of a glass of ice water left on the kitchen table. (b) The amount of money in an interest-bearing bank account into which $50 is deposited. (c) The speed of a constantly decelerating car. (d) The temperature of a piece of steel heated in a furnace and left outside to cool.

Each curve in Figure 11.51 represents the balance in a bank account into which a single deposit was made at time zero. Assuming continuously compounded interest, find: (a) The curve representing the largest initial deposit. (b) The curve representing the largest interest rate. (c) Two curves representing the same initial deposit. (d) Two curves representing the same interest rate.

The slope field for y = 0.5(1 + y)(2 y) is given in Figure 11.52. (a) List equilibrium solutions and state whether each is stable or unstable. (b) Draw solution curves on the slope field through each of the three marked points.

The slope field for a differential equation is given in Figure 11.53. Estimate all equilibrium solutions for this differential equation, and indicate whether each is stable or unstable.

For Exercises 56, sketch solution curves with a variety of initial values for the differential equations. You do not need to find an equation for the solution.

For Exercises 56, sketch solution curves with a variety of initial values for the differential equations. You do not need to find an equation for the solution.

A yam is put in a 200C oven and heats up according to the differential equation dH dt = k(H 200), for k a positive constant.

(a) Find the equilibrium solution to the differential equation dy dt = 0.5y 250. (b) Find the general solution to this differential equation. (c) Sketch the graphs of several solutions to this differential equation, using different initial values for y. (d) Is the equilibrium solution stable or unstable?

(a) Find all equilibrium solutions for the differential equation dy dx = 0.5y(y 4)(2 + y). (b) Draw a slope field and use it to determine whether each equilibrium solution is stable or unstable

(a) A cup of coffee is made with boiling water and stands in a room where the temperature is 20 C. If H(t) is the temperature of the coffee at time t, in minutes, explain what the differential equation dH dt = k(H 20) says in everyday terms. What is the sign of k? (b) Solve this differential equation. If the coffee cools to 90C in 2 minutes, how long will it take to cool to 60C degrees?

In Example 2 on page 612, we saw that it would take about 6 years for 90% of the pollution in Lake Erie to be removed and about 12 years for 99% to be removed. Explain why one time is double the other.

Using the model in the text and the data in Table 11.4 on page 612, find how long it would take for 90% of the pollution to be removed from Lake Michigan and from Lake Ontario, assuming no new pollutants are added. Explain how you can tell which lake will take longer to be purified just by looking at the data in the table.

Use the model in the text and the data in Table 11.4 on page 612 to determine which of the Great Lakes would require the longest time and which would require the shortest time for 80% of the pollution to be removed, assuming no new pollutants are being added. Find the ratio of these two times.

In 2010, the population of India was 1.15 billion people and increasing at a rate proportional to its population. If the population is measured in billions of people and time is measured in years, the constant of proportionality is 0.0135.

Nicotine leaves the body at a rate proportional to the amount present, with constant of proportionality 0.347 if the amount of nicotine is in mg and time is in hours. The amount of nicotine in the body immediately after smoking a cigarette is 0.4 mg.

In 2007, world solar photovoltaic (PV) market installations were 2826 megawatts and were growing exponentially at a continuous rate of 48% per year.5

In 2007, Grinnell Glacier in Glacier National Park covered 142 acres and was estimated to be shrinking exponentially at a continuous rate of 4.3% per year.6

Since 1980, textbook prices have increased at 6.7% per year while inflation has been 3.3% per year.7 Assume both rates are continuous growth rates and let time, t, be in years since the start of 1980. (a) Write a differential equation satisfied by B(t), the price of a textbook at time t. (b) Write a differential equation satisfied by P(t), the price at time t of an item growing at the inflation rate. (c) Solve both differential equations. (d) What is the doubling time of the price of a textbook? (e) What is the doubling time of the price of an item growing according to the inflation rate? (f) How is the ratio of the doubling times related to the ratio of the growth rates? Justify your answer

Write a differential equation whose solution is the temperature as a function of time of a bottle of orange juice taken out of a 40F refrigerator and left in a 65F room. Solve the equation and graph the solution.

A roast is taken from the refrigerator, where the temperature is 40F, and put in a 350F oven. One hour later, the meat thermometer shows a temperature of 90F. If the roast is done when its temperature reaches 140F, what is the total time the roast should be in the oven?

Warfarin is a drug used as an anticoagulant. After administration of the drug is stopped, the quantity remaining in a patients body decreases at a rate proportional to the quantity remaining. The half-life of warfarin in the body is 37 hours. (a) Sketch the quantity, Q, of warfarin in a patients body as a function of the time, t, since stopping administration of the drug. Mark the 37 hours on your graph. (b) Write a differential equation satisfied by Q. (c) How many days does it take for the drug level in the body to be reduced to 25% of the original level?

The rate at which a drug leaves the bloodstream and passes into the urine is proportional to the quantity of the drug in the blood at that time. If an initial dose of Q0 is injected directly into the blood, 20% is left in the blood after 3 hours. (a) Write and solve a differential equation for the quantity, Q, of the drug in the blood after t hours. (b) How much of this drug is in a patients body after 6 hours if the patient is given 100 mg initially?

Oil is pumped continuously from a well at a rate proportional to the amount of oil left in the well. Initially there were 1 million barrels of oil in the well; six years later 500,000 barrels remain. (a) At what rate was the amount of oil in the well decreasing when there were 600,000 barrels remaining? (b) When will there be 50,000 barrels remaining?

The radioactive isotope carbon-14 is present in small quantities in all life forms, and it is constantly replenished until the organism dies, after which it decays to stable carbon-12 at a rate proportional to the amount of carbon-14 present, with a half-life of 5730 years. Suppose C(t) is the amount of carbon-14 present at time t. (a) Find the value of the constant k in the differential equation C = kC. (b) In 1988 three teams of scientists found that the Shroud of Turin, which was reputed to be the burial cloth of Jesus, contained 91% of the amount of carbon-14 contained in freshly made cloth of the same material.8 How old is the Shroud of Turin, according to these data?

The amount of radioactive carbon-14 in a sample is measured using a Geiger counter, which records each disintegration of an atom. Living tissue disintegrates at a rate of about 13.5 atoms per minute per gram of carbon. In 1977 a charcoal fragment found at Stonehenge, England, recorded 8.2 disintegrations per minute per gram of carbon. Assuming that the half-life of carbon-14 is 5730

A detective finds a murder victim at 9 am. The temperature of the body is measured at 90.3F. One hour later, the temperature of the body is 89.0F. The temperature of the room has been maintained at a constant 68F. (a) Assuming the temperature, T , of the body obeys Newtons Law of Cooling, write a differential equation for T . (b) Solve the differential equation to estimate the time the murder occurred.

At 1:00 pm one winter afternoon, there is a power failure at your house in Wisconsin, and your heat does not work without electricity. When the power goes out, it is 68F in your house. At 10:00 pm, it is 57F in the house, and you notice that it is 10F outside. (a) Assuming that the temperature, T , in your home obeys Newtons Law of Cooling, write the differential equation satisfied by T . (b) Solve the differential equation to estimate the temperature in the house when you get up at 7:00 am the next morning. Should you worry about your water pipes freezing? (c) What assumption did you make in part (a) about the temperature outside? Given this (probably incorrect) assumption, would you revise your estimate up or down? Why?

Before Galileo discovered that the speed of a falling body with no air resistance is proportional to the time since it was dropped, he mistakenly conjectured that the speed was proportional to the distance it had fallen. (a) Assume the mistaken conjecture to be true and write an equation relating the distance fallen, D(t), at time t, and its derivative. (b) Using your answer to part (a) and the correct initial conditions, show that D would have to be equal to 0 for all t, and therefore the conjecture must be wrong.

(a) An object is placed in a 68F room. Write a differential equation for H, the temperature of the object at time t. (b) Find the equilibrium solution to the differential equation. Determine from the differential equation whether the equilibrium is stable or unstable. (c) Give the general solution for the differential equation. (d) The temperature of the object is 40F initially and 48F one hour later. Find the temperature of the object after 3 hours

Hydrocodone bitartrate is used as a cough suppressant. After the drug is fully absorbed, the quantity of drug in the body decreases at a rate proportional to the amount left in the body. The half-life of hydrocodone bitartrate in the body is 3.8 hours, and the usual oral dose is 10 mg. (a) Write a differential equation for the quantity, Q, of hydrocodone bitartrate in the body at time t, in hours since the drug was fully absorbed. (b) Find the equilibrium solution of the differential equation. Based on the context, do you expect the equilibrium to be stable or unstable? (c) Solve the differential equation given in part (a). (d) Use the half-life to find the constant of proportionality, k. (e) How much of the 10 mg dose is still in the body after 12 hours?

(a) Let B be the balance at time t of a bank account that earns interest at a rate of r%, compounded continuously. What is the differential equation describing the rate at which the balance changes? What is the constant of proportionality, in terms of r? (b) Find the equilibrium solution to the differential equation. Determine whether the equilibrium is stable or unstable and explain what this means about the bank account. (c) What is the solution to this differential equation? (d) Sketch the graph of B as function of t for an account that starts with $1000 and earns interest at the following rates: (i) 4% (ii) 10% (iii) 15%

The line y = 2 is an equilibrium solution to the differential equation dy/dx = y3 4xy.

The function y = x2 is an equilibrium solution to the differential equation dy/dx = y x2

At time t = 0, a roast is taken out of a 40F refrigerator and put in a 350F oven. If H represents the temperature of the roast at time t minutes after it is put in the oven, we have dH/dt = k(H 40).

A differential equation for a quantity that is decaying exponentially at a continuous rate per unit time.

A differential equation with an equilibrium solution of Q = 500.

A graph of three possible solutions, with initial P-values of 20, 25, and 30, respectively, to a differential equation that has an unstable equilibrium solution at P = 25.