Stats for Exam 1. Letter Score Range Frequency A 90 to B 80 to 89 3 C 70 to 79 4 D 60 to 69 4 F 59 and below 8

Transcription

1 Stats for Exam 1 Letter Score Range Frequency A 90 to B 80 to 89 3 C 70 to 79 4 D 60 to 69 4 F 59 and below 8 High Score 100 two of them 75th percentile 94 Median 81 25th percentile 60 Low Score 29 Mean Score Standard deviation

2 Name: SOLUTIONS Math 132 Exam 1 February 6, 2009 You must show all work to receive full credit; unjustified answers will receive LITTLE TO NO CREDIT! Making it very clear what you are doing will increase your eligibility for partial credit. 1. A certain store determines their retail prices by charging 40% above the wholesale price. Thus, if the wholesale price of an item is $1.00 the store will charge $1.40. (a) (5 points) A particular item s wholesale price is $ What is this item s retail price? Solution: If w denotes wholesale price and r denotes retail price, then r = (1.40) w. Thus, $25 wholesale corresponds to (1.40) 25 = 35. (b) (5 points) A particular item s retail price is $ What is this item s wholesale price? Solution: Using the formula above, we get 63 = 1.40 r so r = = A record company advances $50, for a band to record a CD. It costs the company $1.75 to copy, package and distribute each copy of this CD. The record company receives $7.50 for each CD sold. Let x denote the number of CD s distributed and sold, C(x) denote the total cost to produce x CD s, and R(x) denote the revenue generated by selling x CD s. (a) (3 points) Write down the company s cost function. Solution: Fixed cost of and 1.75 per item, so C(x) = 1.75x (b) (3 points) Write down the company s revenue function. Solution: The company receives 7.50 per item, so the revenue is R(x) = 7.50x. 2

3 (c) (4 points) How many CD s need to be sold for the record company to break even? Solution: The company breaks even when the revenue and the cost are equal. Thus 1.75x = 7.50x = 5.75x x = = They can t actually sell 0.7 CDs so we should take x = The price-demand function for a particular product is p(x) = 200 4x where x is the number of items sold, and p(x) is the price per item when x items are sold. (a) (5 points) Write down the revenue function. Solution: Revenue is price per item times number of items. So R(x) = x p(x) = x (200 4x). (b) (5 points) Find x which will give the maximum revenue. Also, find the maximum revenue. Solution: The revenue function above is quadratic so the maximum occurs at the vertex and the maximum revenue is the vertical coordinate of the vertex. You can find the vertex in a number of ways. One method is to observe the x coordinate of the vertex is halfway between the two zeros. But R(x) = 0 implies x = 0 or 200 4x = 0, so x = 50. The x coordinate of the vertex is then x = = 25. The maximum revenue is then R(25) =

4 4. (10 points) The graph shows y = f(x). Sketch the graph of y = f(x 2) + 1. Solution: y = f(x 2) + 1 is obtained by shifting the given graph up 1 and to the right 2 units. 4

5 5. (10 points) You want to buy a home in seven years. After shopping around you find a bank that offers 4.2% annual interest compounded quarterly. How much money would you need to invest today to accumulate $60, 000 in seven years? Your answer should be correct to the nearest dollar. Solution: Apply the compund interest formula A = P (1 + r m )mt. Here, A = 60000, t = 7, r = and m = 4. So, P = ( = ) (10 points) You invested $ some time ago in a bank account that has been earning 3.5% annual interest. This money is now worth $1, Assuming the interest is compounded annually, how long ago was this money invested? Your answer should be given in years, and should be correct to two digits after the decimal. Solution: Apply the compund interest formula A = P (1 + r m )mt. Here, A = 1200, P = 700, r = and m = 1. So, 1200 = 700 (1.035) t 12 7 = (1.035)t ( ) 12 ln = ln (1.035) t = t ln (1.035) 7 t = ln 12/7 ln =

6 7. (10 points) Find the it x 4 3x 2 12 x Solution: 2 8x+12 is a rational function and x = 4 does not give a zero in the 3x 2 12 denominator. Therefore, the it can be obtained by plugging in x = 4. x 4 3x 2 12 = = 4 36 = (10 points) Find the it x 2 3x 2 12 Solution: This time the denominator is zero so it is necessary to factor. 3x 2 12 = (x 6)(x 2) 3(x + 2)(x 2) = x 6 3 (x + 2). Now, x 2 3x 2 12 x 6 = x 2 3 (x + 2 = (2 + 2) = 4 12 =

7 9. F (x) is defined below. F (x) = { x 2, x 2; x, x > 2. (a) (5 points) Sketch the graph of F (x). Solution: The graph is a parabola to the left of 2 and a line to the right of 2. The strict inequality x > 2 tells us to put a hollow circle at (2, 2). The inequality x 2 tells us to put a solid dot at (2, 4). (b) (5 points) Where is F (x) continuous? Discontinuous? Solution: Each piece y = x and y = x 2 is continuous everywhere, so the only possible discontinuity occurs at x = 2. F (x) will be continuous at x = 2 if the two pieces match up at x = 2. But plugging 2 into y = x gives y = 2 and plugging 2 into y = x 2 gives 4. So, F (x) is discontinuous at x = 2 and continuous everywhere else. 7

8 10. (10 points) Find the it 3x 2 2x + 6 x 1 8x 2 Solution: We can find this it in a number of ways, looking at the graph, plugging in extremely large values of x, etc. The quickest is to use the fact that since the numerator and denominator have the same degree the it will be the ratio of the two leading coefficients, in this case 3x 2 2x + 6 = 3 x 1 8x 2 8 =