 # Pre-Calculus Problems

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## Transcribed Text

QUESTION ONE (a) Use the quadratic formula to determine the roots of the equation: 2x2 + x - 1 = 0. (b) Determine the vertex of the parabola y = 2x2+ x - 1. (Recall, the x value of the vertex is given by x = -b/2a) (c) Use parts a and b to sketch the graph of y = 2x2+ x - 1. You may wish to determine a few other “points that the parabola passes through” to obtain a more accurate graph. QUESTION TWO How high will a baseball that is thrown up into the air at 48 feet per second go if it starts its flight at a height 6 feet above the ground? (Hint: This is another application of the quadratic, recall the x value of the vertex is given by x = -b/2a) QUESTION THREE (a) (b) Solve for x: | x - 1 | = 4 (c) Sketch the graph of the function y = | x - 1 | and use your graph to indicate your solution of part b. QUESTION FOUR (a) Perform the indicated operation and express the result in the form a + bi. (b) Simplify. - i 11 (c) Find all real numbers x and y that satisfy the equation: -3x - 1 + 2yi = 4x – 5i. See example 11, section 5.3 QUESTION FIVE Find the area bounded by the following (make sure that you compute the area): y = -x, the x axis and x = 5 QUESTION SIX A rain gutter is to be made up of rectangular aluminum sheets 12 inches wide by turning up the sides edges 90 degrees. What depth (of the edges) will provide a maximum cross sectional area and thereby provide for the greatest flow of water? (Hint, think of quadratic equations and that you want to maximize the “cross sectional area”.) QUESTION SEVEN The profit P, in dollars, for a company that produces antivirus and systems utilities software is: P = -0.0002x2 +140x – 250,000 where x is the number of units sold. What sales level will yield a maximum profit? QUESTION EIGHT A worker (on earth) dropped a screwdriver from the top of an elevator shaft. Exactly 5 seconds later he heard the sound of the screwdriver hitting the bottom of the shaft. How tall is the elevator shaft? QUESTION NINE This is easier than you think. Try it. A bicycle manufacturer has determined that when x hundred bicycles are built, the average cost per bicycle is given by C(x) = 0.1x2 – 0.7 x + 2.425, where C(x) is in hundreds of dollars. How many bicycles should be built in order to minimize the average cost per bicycle? Some key thoughts : (a) Any function of the form y = ax2 + bx + c where a ≠ 0 is a quadratic. (b) The formula gives you the x value of the vertex. Substitute this value in the equation (of part (a)) to find the y value. (c) To sketch the graph of a quadratic one frequently need only know the vertex (x and y values) and the roots (found by solving ax2 +bx + c = 0 by factoring or by using the quadratic formula).The quadratic formula is: -h-vh-4ne x 2a

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