C. HECKMAN TEST 1A SOLUTIONS 170

Transcription

1 C. HECKMAN TEST 1A SOLUTIONS 170 1) Thornley s Bank of Atlanta offers savings accounts which earn 4.5% per year. You have $00, which you want to invest. a) [10 points] If the bank compounds the interest monthly, how much will you have in ten years assuming you don t deposit or withdraw any money)? Solution: Use the compound interest formula; you need to find A: Grading: Shown above. A = P 1 + n) r nt = ) = $ b) [10 points] If the bank compounds the interest continually, how much will you have in ten years? Solution: You need to use the continual compounding formula here, again looking for A: A = P e rt = 00 e A = $ Grading: Shown above. Common mistakes: Forgetting parentheses when entering the expression into a calculator $09.06). 1

2 c) [15 points] When will you have $600 in your account, if the interest is compounded continually? Solution: Here you are using the continual compounding formula here, except you need to solve for t: A = P e rt 600 = 00 e 0.045t 3 = e 0.045t ln3) = ln e 0.045t) = 0.045t t = ln3) years. Grading: Shown above. 5 points for guessing for the value of t. ) Find all solutions to the following equations. You must solve them algebraically, finding exact answers. a) [15 points] log 4 x ) + log 4 x + 10) = 3 Solution: Combine the logarithms, convert to exponential form, and solve the resulting quadratic equation: log 4 x )x + 10) ) = 3 x )x + 10) = 4 3 = 64 x x + 10x 0 = 64 [+5 points] 0 = x + 8x 84 = x 6)x + 14) [+5 points] This implies that x = 6 or x = 14. However, x = 14 is not a legitimate solution, since you are taking the logarithm of a negative number in the original equation. The only solution is x = 6. Grading for common mistakes: 3 points for including x = 14. b) [15 points] e x + 3e x 10 = 0 Solution: To do this one, make the substitution y = e x. Then you get the equation y + 3y 10 = 0, which factors into y )y + 5) = 0. Thus y = which means x = ln)) or y = 5 and there is no corresponding value for x); the solution is just ln). Grading: +5 points for the substitution, +5 points for factoring, +5 points for finding x. Grading for common mistakes: 3 points for the answers and 5; 3 points for the answers ln) and ln 5); +3 points total) for miscellaneous work.

3 3) [15 points] A scientist is keeping a sample of Copper-64 in a laboratory until 80% of the Copper-64 has decayed radioactively into something else. How long will he have to wait? The half-life of Copper-64 is hours.) Solution: Use the radioactive decay equation; you need to solve for t: ) t/h A 1 = A 0 ) t/ = ) t/1.701 ) 1 t 1 ln0.0) = ln = ln t = ln0.0) ln1/) hours Grading for common mistakes: points for using 0.80 instead of for A A 0 answer: 4.09 hours). 4) [10 points] Find log 7 5) to four decimal places. Solution: log 7 5) = ln5) = or Common logarithms could ln7) also have been used; log5) = as well.) log7) Grading: +5 points for the change-of-base formula, +5 points for calculation. Grading for common mistakes: +3 points total for no work; points for ln7) ln5). 5) [10 points] Find the domain of the function log 8 x x 6). Write your answer in interval notation. Solution: You can only take a logarithm of a positive number, so you must have x x 6 > 0. Note that x x 6 = 0 when x = 3 or, so there are three pieces of the real number line to check. It turns out that x x 6 > 0 when x < or when x > 3 but not when x is between and 3. The domain is thus all real numbers less than or greater than 3, which in interval notation is, ) 3, + ). Grading: +4 points for the inequality x x 6 > 0, +3 points for testing usually using a real number line), +3 points for writing the answer. Grading for common mistakes: points for, ] [3, + ). 3

4 C. HECKMAN TEST 1B SOLUTIONS 170 1) Thornley s Bank of Atlanta offers savings accounts which earn 3.5% per year. You have $300, which you want to invest. a) [10 points] If the bank compounds the interest quarterly, how much will you have in five years assuming you don t deposit or withdraw any money)? Solution: Use the compound interest formula; you need to find A: A = P 1 + n) r nt = ) = $ Grading: Shown above. Grading for common mistakes: points for letting n = 3 answer: $357.01). b) [10 points] If the bank compounds the interest continually, how much will you have in five years? Solution: You need to use the continual compounding formula here, again looking for A: A = P e rt = 300 e A = $

5 c) [15 points] When will you have $500 in your account, if the interest is compounded continually? Solution: Here you are using the continual compounding formula here, except you need to solve for t: A = P e rt 500 = 300 e 0.035t 5 3 = e0.035t ) 5 ln = ln e 0.035t) = 0.035t 3 t = ln5/3) years Grading: Shown above. 5 points for guessing for the value of t, or for not taking logs when solving for t. ) Find all solutions to the following equations. You must solve them algebraically, finding exact answers. a) [15 points] log 3 x 1) + log 3 x + 7) = Solution: Combine the logarithms, convert to exponential form, and solve the resulting quadratic equation: log 3 x 1)x + 7) ) = x 1)x + 7) = 3 = 9 x x + 7x 7 = 9 [+5 points] 0 = x + 6x 16 = x )x + 8) [+5 points] This implies that x = or x = 8. However, x = 8 is not a legitimate solution, since you are taking the logarithm of a negative number in the original equation. The only solution is x =. Grading for common mistakes: 3 points for including x = 8. b) [15 points] e x 5e x + 6 = 0 Solution: To do this one, make the substitution y = e x. Then you get the equation y 5y + 6 = 0, which factors into y )y 3) = 0. Thus y = which means x = ln)) or y = 3 which means x = ln3)); the solutions are ln), ln3). Grading: +5 points for the substitution, +5 points for factoring, +5 points for finding x. Grading for common mistakes: 3 points for the answers and 3; +3 points total) for miscellaneous work.

6 3) [15 points] A scientist is keeping a sample of Lead-1 in a laboratory until 5% of the Lead-1 has decayed radioactively into something else. How long will she have to wait? The half-life of Lead-1 is hours.) Solution: Use the radioactive decay equation; you need to solve for t: ) t/h A 1 = A 0 ) t/ = ) t/ ln0.75) = ln = t ) ln t = ln0.75) ln1/) hours Grading for common mistakes: points for using 0.5 instead of for A A 0 answer: 1. hours). 4) [10 points] Find log 3 5) to four decimal places. Solution: log 3 5) = ln5) = or Common logarithms could ln3) also have been used; log5) = as well.) log3) Grading: +5 points for the change-of-base formula, +5 points for calculation. Grading for common mistakes: +3 points total for no work; points for ln3) ln5). 5) [10 points] Find the domain of the function log 9 4x x ). Write your answer in interval notation. Solution: You can only take a logarithm of a positive number, so you must have 4x x > 0. Note that 4x x = 0 when x = 0 or 4, so there are three pieces of the real number line to check. It turns out that 4x x > 0 only when x is between 0 and 4. The domain is thus all real numbers between 0 and 4, which in interval notation is 0, 4). Grading: +4 points for the inequality 4x x > 0, +3 points for testing usually using a real number line), +3 points for writing the answer. Grading for common mistakes: points for [0, 4]. 3