The management at a factory has determined that the rate of change of the number of units $N$ that a new employee can produce per day after having worked $t$ days is proportional to $N-30$. After 0 days on the job, the employee produces 0 units and after 15 days on the job, the employee produces 15 units.

According to your model, how many days pass before the employee produces 25 units per day?

The current no-smoking regulations in office buildings require workers who smoke to take breaks and leave the building in order to satisfy their habits. A study indicates that such workers average 32 minutes per day taking smoking breaks. The standard deviation is 8 minutes. To help reduce the average break, rooms with powerful exhausts were installed in the buildings. To see whether these rooms serve their designed purpose, a random sample of 110 smokers was taken. The total amount of time away from their desks was measured for 1 day. Test to determine whether there has been a decrease in the mean time away from their desks. Compute the p-value and interpret it relative to the costs of Type I and Type II errors.

Let v(t) be the velocity, in feet per second, of a skydiver at time t seconds, t ≥ 0. After her parachute opens, her velocity satisfies the differential equation dv/dt = -2(v + 17), with initial condition v(0) = - 47. Use separation of variables to find an expression for v in terms of t, where t is measured in seconds.

The management at a factory has determined that the rate of change of the number of units $N$ that a new employee can produce per day after having worked $t$ days is proportional to $N-30$. After 0 days on the job, the employee produces 0 units and after 15 days on the job, the employee produces 15 units.

Write and find the general solution of the differential equation that models this situation.

A population of eight beavers has been introduced into a new wetlands area. Biologists estimate that the maximum population the wetlands can sustain is 60 beavers. After 3 years, the population is 15 beavers. According to a Gompertz growth model, how many beavers will be present in the wetlands after 10 years ?

A differential equation and its slope field are given. Complete by determining the slopes (if possible) in the slope field at the given points. x: -4, -2, 0, 2, 4, 8 y: 2, 0, 4, 4, 6, 8. dy/dx = 2x/y

The drug company Pfizer is considering whether to invest in the development of a new cancer drug. Development will require an initial imvestment of $10 million now; beginning one year from now, the drug will generate annual profits of$4 million for three years.

If the interest rate is 12%, should Pfizer invest in the development of the new drug? Why or why not?

A container of hot liquid is placed in a freezer that is kept at a constant temperature of $20^{\circ} \mathrm{F}$. The initial temperature of the liquid is $160^{\circ} \mathrm{F}$. After 5 minutes, the liquid's temperature is $60^{\circ} \mathrm{F}$.

How much longer will it take for the temperature of the liquid to decrease to $25^{\circ} \mathrm{F}$ ?

A container of hot liquid is placed in a freezer that is kept at a constant temperature of $20^{\circ} \mathrm{F}$. The initial temperature of the liquid is $160^{\circ} \mathrm{F}$. After 5 minutes, the liquid's temperature is $60^{\circ} \mathrm{F}$.

Write an equation for the temperature $y$ of the liquid $t$ minutes after it is placed in the freezer.

In this exercise, use the Gompertz growth model described on the said page to find the growth function, and sketch its graph.

$L=5000 ; y=500$ when $t=0 ; y=625$ when $t=1$

Use integration to find a general solution of the differential equation.

$$\frac{d y}{d x}=5(\sin x) e^{\cos x}$$

When an object is removed from a furnace and placed in an environment with a constant temperature of $80^{\circ} \mathrm{F}$, its core temperature is $1500^{\circ} \mathrm{F}$. One hour after it is removed, the core temperature is $1120^{\circ} \mathrm{F}$.

What is the core temperature of the object 6 hours after it is removed from the furnace?

When an object is removed from a furnace and placed in an environment with a constant temperature of $80^{\circ} \mathrm{F}$, its core temperature is $1500^{\circ} \mathrm{F}$. One hour after it is removed, the core temperature is $1120^{\circ} \mathrm{F}$.

Write an equation for the core temperature $y$ of the object $t$ hours after it is removed from the furnace.

In this exercise, use the Gompertz growth model described on the said page to find the growth function, and sketch its graph.

$L=500 ; y=100$ when $t=0 ; y=150$ when $t=2$