Exam 9: First-Order Differential Equations

The University of XYZ has a goal to increase its endowment from the initial value of $200,000,000, to $300,000,000 over 5 years. If the interest rate earned by the endowment (after expenses) is 5% each year (compounded continuously), and the contributions become available continuously and at a constant rate, how much will they actually have to collect from contributors over those 5 years to meet their goal?

You select an insurance policy for your company that agrees to pay you the depreciated value of a piece of equipment if it is destroyed in an accident, but you have to choose one of two depreciation schedules when you begin the policy. One depreciation schedule is linear and depreciates the value to zero over 10 years. The other depreciation schedule is exponential and depreciates the value at a constant rate of 20% per year. An accident destroys the equipment after 5 years of ownership. Calculate the insurance company's obligation for each plan if the piece of equipment was valued initially at $100,000. Which depreciation schedule would have paid the most? (The cost of the policy is the same, no matter which depreciation schedule you choose.)

A particular investment program involves continuously making small investments adding up to $3600 each year. If the investment pays 8%, and the account started off with an initial balance of $1000, how much will the account be worth in 40 years?

Use your calculator to construct the direction field for the following differential equation. [Use the standard zoom window.]

Identify the equilibrium solutions for , and determine if they are stable or unstable.

The conversion of sucrose (table sugar) to glucose and fructose is first order in the concentration of sucrose, which means that the rate of reaction is proportional to the concentration of the sucrose. The rate of disappearance of sucrose can be expressed as , where c represents the concentration of the sucrose, and k is called the rate constant and is mathematically identical to the negative of the decay constant. If it takes 4 hours for the sucrose concentration to drop by a factor of 6, what is the rate constant?

Find the solution to the following separable differential equation. , a is a constant.

The number of bacteria in a culture increases exponentially with a growth constant of 0.3 hour^-1. How long will it take for the population to increase from 6000 to 30,000 ?

Find all equilibrium points for the following system of equations.

A treasury note that will be worth $100,000 in 10 years currently sells for $81,058. What constant interest rate does that correspond to?

Is the following differential equation separable or not?

The population of New Zealand grew exponentially through the 20th century at a rate of 1.6% year^-1. If the population in 2000 was 3.9 million, when was the population 3.0 million?

Match the appropriate slope field with the differential equation .

At the cafeteria, two identical glasses of juice were poured at the same time and put on the counter waiting for customers to take them. Their temperatures were 33^oF when they were poured, and the cafeteria was a stable 72^oF. One glass was 42^oF when it was taken after 40 seconds. What was the temperature of the other when it was taken after 200 seconds?

Find the solution of the differential equation, y' = -4y, satisfying the initial condition, y(5) = 0.

Suppose the income tax structure is as follows: the first $16,000 is taxed at 10%, the remainder is taxed at 35%. Compute the tax on an income of $40,000. Now suppose that inflation is 5% and you receive a cost of living (5%) raise to $42,000. Compute the tax on this income. To compare the taxes you should adjust the tax for inflation (add 5%).

A stirred tank with volume, V, has a feed stream of concentrated floor cleaner flowing into it at a rate of f . The flow stream has a concentration of c_i. The outlet stream also flows at f but with a concentration of c, the concentration of the floor cleaner solution in the tank. The rate of change of the concentration is proportional to the difference between c_i and c with proportionality constant of f/V: . If the tank volume is 1000 L, the flow rate is 10 L/min, the inlet concentration is 1.0, and the initial concentration of the floor cleaner in the tank is 0.0, how long until the concentration in the tank is 0.7?

Find the solution to the following separable differential equation. , a is a constant

Is the following differential equation separable or not?

Identify the equilibrium solutions for , and determine if they are stable or unstable.