91Ó°ÊÓ

Suppose that \(f(t)\) satisfies the initial-value problem \(y^{\prime}=y^{2}+t y-7, y(0)=2\). Is the graph of \(f(t)\) increasing or decreasing at \(t=0\) ?

A Savings Account A person deposits $$\$ 10,000$$ in a bank account and decides to make additional deposits at the rate of \(A\) dollars per year. The bank compounds interest continuously at the annual rate of \(6 \%\), and the deposits are made continuously into the account. (a) Set up a differential equation that is satisfied by the amount \(f(t)\) in the account at time \(\bar{t}\). (b) Determine \(f(t)\) (as a function of \(A\) ). (c) Determine \(A\) if the initial deposit is to double in 5 years.

Review concepts that are important in this section. In each exercise, sketch the graph of a function with the stated properties. Domain: \(0 \leq t \leq 8 ;(0,6)\) is on the graph; the slope is always negative, the slope becomes more negative as \(t\) increases from 0 to 3 , and the slope becomes less negative as \(t\) increases from 3 to 8 .

A person took out a loan of $$\$ 100,000$$ from a bank that charges \(7.5 \%\) interest compounded continuously. What should be the annual rate of payments if the loan is to be paid in full in exactly 10 years? (Assume that the payments are made continuously throughout the year.)

Solve the following differential equations: $$ y^{\prime}=\left(\frac{t}{y}\right)^{2} e^{t^{3}} $$

In an autocatalytic reaction, one substance is converted into a second substance in such a way that the second substance catalyzes its own formation. This is the process by which trypsinogen is converted into the enzyme trypsin. The reaction starts only in the presence of some trypsin, and each molecule of trypsinogen yields one molecule of trypsin. The rate of formation of trypsin is proportional to the product of the amounts of the two substances present. Set up the differential equation that is satisfied by \(y=f(t)\), the amount (number of molecules) of trypsin present at time \(t .\) Sketch the solution. For what value of \(y\) is the reaction proceeding the fastest? [Note: Letting \(M\) be the total amount of the two substances, the amount of trypsinogen present at time \(t\) is \(M-f(t) .]\)

Let \(q=f(p)\) be the demand function for a certain commodity, where \(q\) is the demand quantity and \(p\) the price of 1 unit. In Section 5.3, we defined the elasticity of demand as $$ E(p)=\frac{-p f^{\prime}(p)}{f(p)} $$ (a) Find a differential equation satisfied by the demand function if the elasticity of demand is a linear function of price given by \(E(p)=p+1\). (b) Find the demand function in part (a), given \(f(1)=100\).

A body was found in a room when the room's temperature was \(70^{\circ} \mathrm{F}\). Let \(f(t)\) denote the temperature of the body \(t\) hours from the time of death. According to Newton's law of cooling, \(f\) satisfies a differential equation of the form $$ y^{\prime}=k(T-y) $$ (a) Find \(T\). (b) After several measurements of the body's temperature, it was determined that when the temperature of the body was 80 degrees it was decreasing at the rate of 5 degrees per hour. Find \(k\). (c) Suppose that at the time of death the body's temperature was about normal, say \(98^{\circ} \mathrm{F}\). Determine \(f(t)\). (d) When the body was discovered, its temperature was \(85^{\circ} \mathrm{F}\). Determine how long ago the person died.

The fish population in a pond with carrying capacity 1000 is modeled by the logistic equation $$ \frac{d N}{d t}=\frac{.4}{1000} N(1000-N) $$ Here, \(N(t)\) denotes the number of fish at time \(t\) in years. When the number of fish reached 250, the owner of the pond decided to remove 50 fish per year. (a) Modify the differential equation to model the population of fish from the time it reached 250 . (b) Plot several solution curves of the new equation, including the solution curve with \(N(0)=250\). (c) Is the practice of catching 50 fish per year sustainable or will it deplete the fish population in the pond? Will the size of the fish population ever come close to the carrying capacity of the pond?

Mothballs tend to evaporate at a rate proportional to their surface area. If \(V\) is the volume of a mothball, then its surface area is roughly a constant times \(V^{2 / 3}\). So the mothball's volume decreases at a rate proportional to \(V^{2 / 3}\). Suppose that initially a mothball has a volume of 27 cubic centimeters and 4 weeks later has a volume of \(15.625\) cubic centimeters. Construct and solve a differential equation satisfied by the volume at time \(t\). Then, determine if and when the mothball will vanish \((V=0)\)