Differential Equations — Quiz 3

Questions in this quiz (9)

1. A bacteria culture begins with $$2$$ cells and grows to $$16$$ cells after one week. Assuming the rate of growth is proportional to the number of cells, how many weeks will it take for the culture to reach $$2000$$ cells?

   • $$3.5$$ weeks
   • $$3.2$$ weeks
   • $$3.8$$ weeks
   • $$4.1$$ weeks

2. A population of rabbits grows at a rate proportional to both the population and the remaining capacity of $$80$$ rabbits. If there were $$12$$ rabbits two months ago and $$40$$ rabbits now, how many rabbits will there be in one month?

   • $$52$$
   • $$55$$
   • $$55$$
   • $$60$$

3. Newton's Law of Cooling states that the rate at which an object cools is proportional to the difference between its temperature and the ambient temperature. A cup of tea was $$180^\circ\text{F}$$ ten minutes ago, the room temperature is $$70^\circ\text{F}$$, and the tea is now $$150^\circ\text{F}$$. What will the temperature be in $$20$$ minutes?

   • $$120^\circ\text{F}$$
   • $$125^\circ\text{F}$$
   • $$130^\circ\text{F}$$
   • $$135^\circ\text{F}$$

4. The population of a species of fish in a lake grows according to a logistic model proportional to $$500-P(t)$$, where $$P$$ is in thousands. At $$t=0$$ the population was $$100{,}000$$ and after $$3$$ years it was $$250{,}000$$. In what year will the population reach $$400{,}000$$?

   • Year $$4$$
   • Year $$5$$
   • Year $$7$$
   • Year $$6$$

5. A stone is thrown straight up from the ground. What should the initial velocity be if you want the stone to reach a height of $$100$$ feet?

   • $$50\text{ ft/sec}$$
   • $$80\text{ ft/sec}$$
   • $$92\text{ ft/sec}$$
   • $$96\text{ ft/sec}$$

6. According to Newton's law of cooling, $$\frac{dT}{dt}=-k(T-15)$$. If a body cools from $$40^\circ C$$ to $$32^\circ C$$ in one hour, find $$T(t)$$.

   • $$T=15+25e^{-0.223t}$$
   • $$T=40e^{-0.223t}$$
   • $$T=15+17e^{-0.223t}$$
   • $$T=25e^{-0.223t}+15$$

7. A cup of tea at temperature $$160^\circ F$$ is placed in a room at $$70^\circ F$$. The differential equation is $$\frac{dy}{dt}=-0.09(y-70)$$. Approximately how long does it take the temperature to drop to $$80^\circ F$$?

   • 16 minutes
   • 18 minutes
   • 18 minutes
   • 22 minutes

8. Which statements characterize logistic growth with limiting value $$M$$ and initial value less than $$M/2$$?

   • I only
   • II only
   • I and II only
   • I, II, and III

9. The table shows selected values of the derivative for a differentiable function $$f.$$Given that $$f(3)=100,$$ use Euler's method with a step size of $$2$$ to estimate $$f(7).$$

   • $$102.5$$
   • $$103$$
   • $$104$$
   • $$104.5$$