 # Pre-Calculus Problems

## Transcribed Text

Question 1 (a) Find all solutions to the equation: x - 7 x + 12 = 0. Check your solutions. (b) Find all solutions to the equation: . Check your solutions. Question 2 Use the quadratic formula to solve the following. Denote whether the roots are Rational or Irrational. Check your solutions. (a) 2x2 + 1 = 5x (b) In addition, sketch the graph of the function y = 2x2 - 5x + 1 using the roots you found in part A and the vertex. Question 3 Find the asymptotes, the intercepts and sketch the following functions. A computer sketch is not sufficient. You must explain. (a) (b) Question 4 A rain gutter is to be made up of rectangular aluminum sheets 12 inches wide by turning up the side edges 90 degrees. What depth (of the edges) will provide a maximum cross sectional area and thereby provide for the greatest flow of water? Question 5 Sketch the graphs of the following functions on the same set of axes: Question 6 Suppose you deposited \$5000 in a savings account with an annual rate of interest of 3% compounded continuously. How much money will be in the account in 10 years? Question 7 (a) Express 4 1 log 2 2  in exponential form. (b) Solve for x: 4 log x 2 (c) Determine the exact value of ln e5 . (Do not give a calculator estimate.) Question 8 A person deposits \$3,000 in a bank account which pays 3% annual interest compounded continuously. How many years will it take for the amount of money in the account to double? Use the below process to determine an exact solution and the check your solution using the estimate of the “law of 72.” /* EXAMPLE FOR QUESTION 8... A person deposits \$1,000 in a bank account which pays 8% annual interest compounded continuously. How many years will it take for the amount of money in the account to double. The mathematical model of this problem is A = Pe.08n. In this case, P = \$1,000 and we want to find n when A = \$2,000. 2000 = 1000 e.08n divide both side by 1000 2 = e.08n take the natural log (ln) of both sides. ln 2 = ln(e.08n) ln 2 = .08n ln e Simplify using ln e = 1 (Do you see why we took the ln of both sides as opposed to log10 of both sides) ln 2 = .08n divide both sides by .08 so that n = ln2 .08 .6931 .08 8.66 So it takes about 8.6 years (or 8 years and 8 months) for the money in the account to double. ... EXAMPLE FOR QUESTION 8 */ Question 9 A lake is formed with a newly constructed dam. It is stocked with1,000 fish. The fish population is expected to increase according to the formula 1.35 30 1 29 N x e where N is the number of fish in thousands expected after t years. The lake will be open to fishing when the number of fish reaches 20,000. How many years to the nearest year will this take? Bonus Questions (a) A small city contains 2500 people. If the population doubles every 25 years, how large will the population be in 225 years? (b) Simplify the following: 2 x -x x -x x -x e + e e - e e - e (c) A fax machine is purchased for \$5,800. Its value each year is about 80% of the value of the preceding year. So after t years the value, in dollars, of the fax machine, V(t), is given by the exponential function: V(t) = 5800 (0.8)t . i. Give a sketch of the graph of the function V(t). Your graph can be a “rough draft”. ii. Determine the value of the fax machine in years 0, 1 and 4. to the nearest tenth. iii. Assume that the company decides to replace the machine when the machines values reduces to \$500. In how many years will the machine be replaced? iv. (d) Solve: 2 x + 1 2 8 for x. Check your solution.

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