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

Problem 1. Given the demand equation $$p=4200-1.5x^2$$ and the supply equation $$p=x^2+200$$, find the producer surplus when the market is in equilibrium. Round your answer to the nearest whole number.

(A) $$64000$$ $$\quad\quad$$ (B) $$128000$$ $$\quad\quad$$ (C) $$42667$$ $$\quad\quad$$ (D) $$85333$$ $$\quad\quad$$ (E) $$106667$$

Problem 2. If the interest rate is $$7\%$$ compounded monthly, how much will $$\8,000$$ accumulate to after $$6$$ years?

(A) $$\12,175.79$$ $$\quad\quad$$ (B) $$\12,160.84$$ $$\quad\quad$$ (C) $$\11,360.00$$ $$\quad\quad$$ (D) $$\11,428.83$$ $$\quad\quad$$ (E) $$\10,690.20$$

Problem 3. The table below shows the number of cell phone subscriptions in the U.S. in millions for the years from $$2003$$ to $$2009$$. Using the year $$2003$$ as the reference year (year zero), find the exponential function that best fits the data, and from that function estimate the number of subscriptions in $$2012$$. $\begin{array}{|c|c|}\hline \mbox{Year}& \mbox{Millions of subscriptions}\\ \hline 2003 & 16.00\\ \hline 2005 & 33.80\\ \hline 2006 & 44.10\\ \hline 2007 & 55.30\\ \hline 2009 & 86.05\\ \hline\end{array}$ (A) $$220.25$$ million $$\quad\quad$$ (B) $$215.13$$ million $$\quad\quad$$ (C) $$212.45$$ million $$\quad\quad$$ (D) $$171.17$$ million $$\quad\quad$$ (E) $$116.55$$ million

Problem 4. If $$s(t)=e^t-t\ln t-3t$$ represents the position of a particle at time $$t$$, find $$a(t)$$, the acceleration of the particle at time $$t=3$$.

(A) $$13.8883$$ $$\quad\quad$$ (B) $$19.7522$$ $$\quad\quad$$ (C) $$14.9869$$ $$\quad\quad$$ (D) $$17.3321$$ $$\quad\quad$$ (E) $$18.9831$$

Problem 5. Given the marginal cost function $$C'(x)=99x^2-24x$$, find the cost of producing $$6$$ items if the fixed cost is $$\1500$$.

(A) $$\6696$$ $$\quad\quad$$ (B) $$\3420$$ $$\quad\quad$$ (C) $$\8196$$ $$\quad\quad$$ (D) $$\4920$$ $$\quad\quad$$ (E) $$\5196$$

Problem 6. Find the area bounded by $$y=x^3-4x^2+1$$ and $$y=x-3$$.

(A) $$\frac{253}{12}\approx 21.08$$ $$\quad\quad$$ (B) $$\frac{-125}{12}\approx -10.42$$ $$\quad\quad$$ (C) $$\frac{211}{12}\approx 17.58$$ $$\quad\quad$$
(D) $$\frac{125}{12}\approx 10.42$$ $$\quad\quad$$ (E) $$\frac{157}{12}\approx 13.08$$

Problem 7. If a company sells an item for $$p=75-0.01x$$ dollars each, and the cost of manufacturing $$x$$ items is $$C(x)=1850+28x-x^2+0.001x^3$$, find the production level which maximizes the profit. Round your answer to the nearest whole number.

(A) $$710$$ $$\quad\quad$$ (B) $$652$$ $$\quad\quad$$ (C) $$844$$ $$\quad\quad$$ (D) $$657$$ $$\quad\quad$$ (E) $$683$$

Problem 8. If $$\5000$$ is invested at the interest rate of $$4.5\%$$ compounded continuously, how long will it take for the amount to grow to $$\25,000$$?

(A) $$33.27$$ years $$\quad\quad$$ (B) $$37.57$$ years $$\quad\quad$$ (C) $$34.77$$ years $$\quad\quad$$ (D) $$36.87$$ years $$\quad\quad$$ (E) $$35.77$$ years

Problem 9. For which $$x$$-values does the graph of $$y=e^{3x^3-2x^2-4x}$$ have a horizontal tangent line?

(A) $$x=1.215$$ and $$x=-0.549$$ $$\quad\quad$$ (B) $$x=1.535$$ $$\quad\quad$$ (C) $$x=-0.481$$ and $$x=0.925$$
(D) $$x=-0.869$$, $$x=0$$, and $$x=1.535$$ $$\quad\quad$$ (E) $$x=-0.758$$ and $$x=-0.208$$

Problem 10. The half-life of a radioactive substance is $$2$$ years. How many years does it take for $$4$$ grams of the substance to decay to $$0.6$$ grams? Round off your answer to two places after the decimal.

(A) $$5.47$$ years $$\quad\quad$$ (B) $$0.347$$ years $$\quad\quad$$ (C) $$2.21$$ years $$\quad\quad$$ (D) $$3.08$$ years $$\quad\quad$$ (E) $$4.27$$ years