Applications of Integration — Quiz 3

Questions in this quiz (10)

1. A ball is thrown upward from the ground with an initial velocity of $$80$$ ft/sec. Given that $$a(t)=-32\ \text{ft/sec}^2$$, the ball's average velocity during the first $$2$$ seconds is:

   • $$16$$ ft/sec
   • $$32$$ ft/sec
   • $$48$$ ft/sec
   • $$64$$ ft/sec

2. The center of a town, assumed to be circular, lies along a straight river that runs along the diameter of the circle. The radius of the town is $$4$$ miles. The density of the population, in thousands of people per square mile, is given by $$f(x)=10-x$$ at a distance $$x$$ miles from the river. The population of the town, in thousands of people, is given by the integral:

   • $$\int_0^4(10-x)dx$$
   • $$2\int_0^4(10-x)\sqrt{16-x^2}dx$$
   • $$4\int_0^4(10-x)\sqrt{16-x^2}dx$$
   • $$\int_0^42\pi x(10-x)dx$$

3. Assume that the density of cyclists, measured in number of cyclists per mile, during a race along a $$15$$-mile road is given by $$f(x)$$, where $$x$$ is the distance, in miles, from the starting line. Which of the following gives the number of cyclists on the road from the starting line to a point $$x$$ miles from the start?

   • $$\int_0^xf(t)dt$$
   • $$\int_x^{15}f(t)dt$$
   • $$\int_0^{15}f(x)dx$$
   • $$\sum_{k=1}^nf(x_k)\Delta x$$

4. A motorcycle accelerates from $$0$$ to $$50$$ mph in $$8$$ seconds, with constant acceleration. Note that $$50$$ mph $$=73.3$$ ft/sec. The acceleration (in ft/sec$$^2$$) is:

   • $$7.16$$
   • $$8.25$$
   • $$9.16$$
   • $$10.25$$

5. Suppose the current population of a country is $$5$$ million and the population $$t$$ years from now is estimated to be $$P(t)=5e^{0.02t}$$ million people. On the basis of this assumption, the average population of the country, in millions, over the next $$20$$ years will be approximately:

   • $$5.74$$
   • $$6.15$$
   • $$6.60$$
   • $$7.42$$

6. chemical is leaking from a container at the rate$$L(t)=800e^{-0.4t}$$ liters per hour, where $$t$$ is measured in hours. The total number of liters leaked during the first $$6$$ hours is approximately:

   • $$1675$$
   • $$1819$$
   • $$1850$$
   • $$2000$$

7. A rumor spreads through a town at the rate$$R(t)=t^2+6t$$ new people per day $$t$$ days after it was first heard. Approximately how many people hear the rumor during the third week from the $$14$$th to the $$21$$st day after it was first heard?

   • $$1043$$
   • $$1519$$
   • $$1987$$
   • $$\frac{8722}{3}$$

8. A park opens at $$9$$ A.M. and visitors arrive at a rate of$$R(t)=12+30t$$ people per hour, where $$t$$ represents the number of hours the park has been open. Assuming no one leaves before noon, at what time will there be $$120$$ people in the park?

   • $$9:28$$ A.M
   • $$10:00$$
   • $$10:15$$
   • $$10:30$$

9. Suppose the amount of a medication in a patient's bloodstream $$t$$ hours after injection is$$A(t)=\frac{40}{(t+2)^2}$$ mg. The average amount of the medication in the bloodstream during the first $$3$$ hours is:

   • $$4.0$$ mg
   • $$5.5$$ mg
   • $$6.8$$ mg
   • $$8.0$$ mg

10. What is the total area bounded by the curve$$f(x)=x^3-5x^2+4x$$ and the $$x$$-axis on the interval $$0\le x\le4$$?

    • $$2.67$$
    • $$3.33$$
    • $$4$$
    • $$11.83$$