Math141Sols2F11

Math141Sols2F11 - MIDTERM 2 SOLUTIONS (CHAPTERS 2 AND 3)...

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MIDTERM 2 SOLUTIONS (CHAPTERS 2 AND 3) MATH 141 – FALL 2011 – KUNIYUKI 150 POINTS TOTAL: 37 FOR PART 1, AND 113 FOR PART 2 Show all work, simplify as appropriate, and use “good form and procedure” (as in class). Box in your final answers! No notes or books allowed. PART 1: SCIENTIFIC CALCULATORS ALLOWED! (37 POINTS TOTAL) 1) A profit function for a statue-making company is given by: px () = ± 200 x 2 + 1200 x ± 200 (in dollars), where x is the number of statues produced, and x ± 0 . (9 points total) a) Use a formula we used in class to find the production level (i.e., the number of statues produced) that will lead to the maximum profit. The graph of p versus x is a parabola opening downward. We want the x -coordinate corresponding to the vertex (i.e., the maximum point) of the parabola. x = ± b 2 a = ± 1200 2 ± 200 = 3 statues b) What is the maximum profit? The corresponding maximum profit is given by: p 3 = ± 200 3 2 + 1200 3 ± 200 = $1600 Note : The graph of p against x is below.
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2) Consider ft () = t 3 ± 7 t 2 + 17 t ± 14 . Hint: One of the zeros is 2. (16 points) a) Write the two other complex zeros of f in simplest, standard form. Box in your answers! (13 points) By the Factor Theorem, because 2 is a zero, t ± 2 is a factor of . Use Synthetic Division to check this and to help us factor . Therefore, = t ± 2 t 2 ± 5 t + 7 . We want to find the zeros (roots) of the quadratic factor, t 2 ± 5 t + 7. We have: a = 1, b = ± 5, c = 7 . Observe that the discriminant of the quadratic factor is: b 2 ± 4 ac = ± 5 2 ± 41 7 = 25 ± 28 = ± 3 This is not a nice square, so traditional factoring will not work. We can find the zeros (roots) using the Quadratic Formula (QF). t = ± b ± 2 ± 4 ac 2 a = ±± 5 ± 3 21 ² We already found the discriminant. = 5 ± i 3 2 = 5 2 ± 2 i Remember that we already verified that 2 was a zero.
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b) Write the polynomial ft () as a product of three linear factors over ± , the set of complex numbers. We basically want the Linear Factorization Theorem (LFT) Form of the factorization. (3 points) = t ± 2 t ± 5 2 + 2 i ² ³ ´ µ · ¸ ¹ º º » ¼ ½ ½ t ± 5 2 ± 2 i ² ³ ´ µ · ¸ ¹ º º » ¼ ½ ½ , or t ± 2 t ± 5 2 ± 2 i ¸ ¹ º º » ¼ ½ ½ t ± 5 2 + 2 i ¸ ¹ º º » ¼ ½ ½ 3) You deposit $7000 into an account that earns interest by continuous compounding at 4% per year. Assuming that there are no further deposits or withdrawals, in how many years will the account have $50,000? Give both an exact answer (which may look ugly) and an approximate answer rounded off to the nearest tenth of a year. (8 points) Model: = Pe rt , where is the amount in the account in t years. Solve for t : 50,000 = 7000 e 0.04 t 50,000 7000 = e 0.04 t 50 7 = e 0.04 t ln 50 7 ± ² ³ ´ µ = ln e 0.04 t ln 50 7 ± ² ³ ´ µ = 0.04 t t = ln 50 7 ± ² ³ ´ µ 0.04 years or 25ln 50 7 ± ² ³ ´ µ years exactly · 49.2 years approximately 4) Approximate log 8 179 to four decimal places. Use a formula we have discussed
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Math141Sols2F11 - MIDTERM 2 SOLUTIONS (CHAPTERS 2 AND 3)...

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