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## MTH 2205: Final Practice 1 (Part 2: Calculators are allowed)

Problem 1. The position function of a particle is given by $$s=t^3-1.5t^2-2t$$, for $$t\geq 0$$. At what value of $$t$$ does the particle reach a velocity of $$166$$ m/sec?

(A) $$t=7$$ sec $$\quad\quad$$ (B) $$t=8$$ sec $$\quad\quad$$ (C) $$t=5$$ sec $$\quad\quad$$ (D) $$t=3$$ sec $$\quad\quad$$ (E) $$t=12$$ sec

Problem 2. The function $$y=\frac{3x^2+x-2}{e^x}$$ has a horizontal tangent line when:

(A) $$x=-3$$ and $$x=1.125$$ $$\quad\quad$$ (B) $$x=-0.468$$ and $$x=2.135$$ $$\quad\quad$$ (C) $$x=-1$$ and $$x=0.667$$
(D) $$x=-0.415$$ and $$x=2.278$$ $$\quad\quad$$ (E) Never

Problem 3. The total cost $$C(x)$$, in dollars, of producing $$x$$ items is given by $C(x)=0.01x^3-0.6x^2+13x.$ What is the maximum profit if each item is sold for $$\6$$? (Assume that everything produced is sold.)

(A) $$\63.03$$ $$\quad\quad$$ (B) $$\23.03$$ $$\quad\quad$$ (C) $$\58.56$$ $$\quad\quad$$ (D) $$\17.82$$ $$\quad\quad$$ (E) There is no maximum profit.

Problem 4. The half-life of a radioactive substance is $$100$$ years. How many years does it take until only $$15\%$$ of the original amount remains?

(A) $$273.7$$ $$\quad\quad$$ (B) $$135.0$$ $$\quad\quad$$ (C) $$282.9$$ $$\quad\quad$$ (D) $$723.5$$ $$\quad\quad$$ (E) $$215.1$$

Problem 5. The population of a city was $$100,000$$ on January 1, 2016 and is growing at a continuous yearly growth rate of $$4.5\%$$. In what year will the population reach $$200,000$$?

(A) $$2028$$ $$\quad\quad$$ (B) $$2031$$ $$\quad\quad$$ (C) $$2034$$ $$\quad\quad$$ (D) $$2037$$ $$\quad\quad$$ (E) $$2040$$

Problem 6. If interest is charged at a nominal rate of $$15.8\%$$ compounded daily (365 days in a year), how much will $$\10,000$$ accumulate to after $$2$$ years? Answer to the nearest dollar.

(A) $$\13160$$ $$\quad\quad$$ (B) $$\10009$$ $$\quad\quad$$ (C) $$\13715$$ $$\quad\quad$$ (D) $$\14102$$ $$\quad\quad$$ (E) $$\12876$$

Problem 7. The table below shows the monthly text messages in billions for the years $$2010$$ to $$2015$$. $\begin{array}{|c|c|}\hline \mbox{Year}& \mbox{Monthly Text Messages (billions)}\\ \hline 2010 & 0.9\\ \hline 2011 & 1.0\\ \hline 2012 & 2.1\\ \hline 2013 & 8.3\\ \hline 2014 & 14.2\\ \hline 2015 & 28.9\\ \hline\end{array}$ Using the year $$2010$$ as the reference year (year zero), find the exponential function that best fits the data, and from that function estimate the number of text messages in $$2017$$.

(A) $$33.1$$ billion $$\quad\quad$$ (B) $$42.4$$ billion $$\quad\quad$$ (C) $$127.5$$ billion $$\quad\quad$$ (D) $$133.2$$ billion $$\quad\quad$$ (E) $$141.6$$ billion

Problem 8. Given the demand curve $$p=35-x^2$$ and the supply curve $$p=3+x^2$$, find the producer surplus when the market is in equilibrium.

(A) $$21.4$$ $$\quad\quad$$ (B) $$42.7$$ $$\quad\quad$$ (C) $$46.4$$ $$\quad\quad$$ (D) $$76.1$$ $$\quad\quad$$ (E) $$91.1$$

Problem 9. The cost function is given by $$C(x)=2x^2-3x+5$$, where $$x$$ is the number of items produced. For what value of $$x$$ is the average cost function minimized?

(A) $$x=1.50$$ $$\quad\quad$$ (B) $$x=5.00$$ $$\quad\quad$$ (C) $$x=1.73$$ $$\quad\quad$$ (D) $$x=1.39$$ $$\quad\quad$$ (E) $$x=1.58$$

Problem 10. What is the area enclosed by the graphs of $$y=x^3-8x^2+18x-5$$ and $$y=x+5$$, shown below? The curves intersect at $$(1,6)$$, $$(2,7)$$, and $$(5,10)$$.

(A) $$10.667$$ $$\quad\quad$$ (B) $$11.833$$ $$\quad\quad$$ (C) $$14.583$$ $$\quad\quad$$ (D) $$21.333$$ $$\quad\quad$$ (E) $$32$$