Differential Equations — Quiz 2

Questions in this quiz (10)

1. If a radioactive substance has a half-life of $$12000$$ years, find an equation that represents the amount remaining after time $$t$$ if you begin with $$M$$ grams.

   • $$M(t)=Me^{-12000t}$$
   • $$M(t)=M\left(\frac12\right)^{t/12000}$$
   • $$M(t)=M2^{-12000t}$$
   • $$M(t)=M\left(\frac12\right)^{12000t}$$

2. A population grows at a rate proportional to its size. Initially, there are $$500$$ individuals. After $$3$$ years, the population grows to $$1500$$. How long will it take for the population to reach $$6000$$?

   • $$6$$ years
   • $$9$$ years
   • $$12$$ years
   • $$15$$ years

3. A species of fish grows exponentially. There are $$200$$ fish initially, and after $$4$$ years, there are $$800$$ fish. How many fish were present after $$2$$ years?

   • $$400$$
   • $$450$$
   • $$500$$
   • $$400$$

4. A new smartphone is released, and the rate of sales is proportional to both the number of phones already sold and the number of potential buyers remaining. The target audience is $$5$$ million people. If $$800000$$ phones were sold by January $$1$$ and $$4$$ million phones were sold by March $$1$$, on what day did the company sell to exactly half of its target audience?

   • January $$25$$
   • February $$1$$
   • February $$1$$
   • February $$18$$

5. The carrying capacity for a fish population in a lake is $$3000$$, and the rate of increase is proportional to both the population $$n$$ and $$3000-n$$. If there were $$800$$ fish one month ago and $$1200$$ fish now, how many months will it take for the population to reach $$2700$$?

   • $$3.5$$ months
   • $$4.5$$ months
   • $$5$$ months
   • $$4$$ months

6. Assume the rate of people entering a concert hall is proportional to both the number already seated and the number of empty seats. The hall has a capacity of $$20000$$. One hour before the show, $$20\%$$ of seats are filled. At showtime, $$70\%$$ are filled. What percentage of seats are filled $$30$$ minutes before showtime?

   • $$45\%$$
   • $$40\%$$
   • $$50\%$$
   • $$55\%$$

7. For what value of $$y$$ is $$\frac{dy}{dt}=ky\left(1-\frac{y}{500}\right)$$ increasing the fastest?

   • $$100$$
   • $$200$$
   • $$250$$
   • $$400$$

8. If $$f''(x)=4x-2$$, $$f'(1)=3$$, and $$f(0)=2$$, find $$f(x)$$.

   • $$\frac{2}{3}x^3-x^2+3x+2$$
   • $$\frac{2}{3}x^3-x^2+2x+2$$
   • $$\frac{4}{3}x^3-x^2+3x+2$$
   • $$\frac{2}{3}x^3-2x^2+3x+2$$

9. Which conic section is the solution to the differential equation $$\frac{dy}{dx}=\frac{12-2x}{8y}$$?

   • Circle
   • Ellipse
   • Parabola
   • Hyperbola

10. What is the position function of a particle moving along the x-axis with velocity $$v(t)=2t^3+3t+1$$ if the particle passes through the origin when $$t=1$$?

    • $$\frac12 t^4+\frac32 t^2+t-3$$
    • $$\frac12 t^4+\frac32 t^2+t-2$$
    • $$t^4+\frac32 t^2+t-3$$
    • $$\frac12 t^4+\frac32 t^2+t-3$$