MATH 2070 Test 3 (Sections , , & )

Transcription

1 Multiple Choice: Use a #2 pencil and completely fill in each bubble on your scantron to indicate the answer to each question. Each question has one correct answer. If you indicate more than one answer, or leave a blank, the question will be marked as incorrect. In this section there are 16 multiple choice questions. Each question is worth 3 points unless otherwise indicated for a total of 46 points. For future reference, circle your answers on this test paper as you will not receive your Scantron back with your test. According to market analysis D( c, p) ice cream bars will be demanded monthly by consumers when the ice cream bar has c calories and sells at a price of p dollars per bar. Use this information to answer the next three questions. 1. What is the demand at point B? a. 412,000 b. 405,000 c. 393,000 d. 380, From point C, which of the following would cause the greatest decrease in demand? a. decreasing the calories by 10 b. increasing the calories by 10 c. decreasing the price by $0.10 d. increasing the price by $ What is the correct classification of points A, B, and C? a. A: relative minimum, B: saddle point, C: relative maximum b. A: relative maximum, B: not a critical point, C: not a critical point c. A: not a critical point, B: saddle point, C: relative maximum d. A: relative maximum, B: saddle point, C: not a critical point P a g e 5 13

2 The table below gives the wind chill C(t,w) in degrees Fahrenheit, where w is the wind speed in miles per hour and t is the air temperature in degrees Fahrenheit. t w Use this context and table to answer the next four questions. 4. Find the function for the linear cross sectional model (equation only) for the wind chill as a function of air temperature when the wind speed is 25 miles per hour. a. C( t, 25) 1.497t b. C( t, 25) 0.667t c. C( t,25) 0.813t d. C( t,25) 1.073t To estimate the wind chill when the air temperature is 30 degrees Fahrenheit and the wind speed is 17 miles per hour you would first need to. a. use a row of data to find the cross-sectional model C(30, w ). b. use a column of data to find the cross-sectional model C(30, w ). c. use a row of data to find the cross-sectional model C( t,17). d. use a column of data to find the cross-sectional model C( t,17). 6. Given the cross-sectional model C(35, w) 45.64(0.93) w, find a. 3.6 b c d dc(35, w) dw dc(35, w) 7. What are the units on? (1 pt) dw w18 a. miles per hour per of wind chill b. of air temperature per of wind chill c. of wind chill per of air temperature d. of wind chill per mile per hour w18. P a g e 6 13

3 TKK Products manufactures 50-, 60-, 75-, and 100-watt electric light bulbs. Laboratory tests show that the lives of these light bulbs are normally distributed with a mean of 750 hr and a standard deviation of 75 hr. Use this context to answer the next three questions. 8. What is the probability that a randomly selected TKK light bulb will burn for less than 700 hours? a b c d At what lifetime was the rate of change of the probability density function for the lives of these light bulbs a maximum? a. 825 hours b. 775 hours c. 675 hours d. 750 hours 10. According to the Empirical Rule, approximately what percentage of light bulbs will last between 675 hours and 975 hours? a. 81.5% b. 82% c % d. 84% 11. From point A, when is K(s, w) increasing most rapidly? a. when s decreases b. when w decreases c. when s increases d. when w increases A P a g e 7 13

4 2 2 P( a, n) 5a 3n 48a 4n 2an 290 million dollars gives the profit of a one-product company where a thousand dollars is the amount spent on advertising and n thousand units of the product are sold. Check: P(6, 2) 402 Use this context to answer the next four questions. 12. Complete the interpretation of P n (6.3,2.1) 4. When a one-product company spends 6.3 thousand dollars on advertising and sells 2.1 thousand units of their product,. a. their profit is decreasing by 4 million dollars per thousand units sold. b. the amount spent on advertising is decreasing by 4 thousand dollars per million dollars of profit. c..their profit is decreasing by 4 million dollars per thousand dollars spent on advertising d. the number of units sold is decreasing by 4 thousand units per million dollars of profit. 13. What system of equations would have to be solved to determine the critical point of P( a, n )? a. 10a 2n 48 2a 6n 4 b. 10a 2n 4 2a 6n 48 c. 10a 2n 4 2a 6n 48 d. 10a 2n 48 2a 6n A graph of the P = 400 million dollar contour curve is shown. Which of the following is a point on the contour curve? a. (6, ) b. (6, ) c. (6, 2.387) d. (6, 2.691) 15. Find the slope of the tangent line at the point (4, 1.721). a b c d P a g e 8 13

5 16. Determine f xy if 5 f ( x, y) y ln( x) 2 y x e x. 2 a. 2 y ln(2) f x x xy b. y f x 2 ln(2) 5e xy 2 5x 1 fxy 2 ln(2) x 5e x c. y 1 2 ln(2) x y d. f xy 5x Check your Scantron now to make sure it will successfully run. If it does, you will earn one point. When you are not working on the multiple choice portion of the test, turn your Scantron over so that it cannot be read by others in the room. P a g e 9 13

6 Free Response: RE-READ the directions at the beginning of the test. Then read each question carefully. Provide only one clearly indicated answer to each question. If your answer is illegible, it will be graded as incorrect. Show all work. The free response portion is 53% of your test grade. When possible, set up the specific mathematical notation that is being evaluated to obtain your answer. No credit will be awarded for simply copying generic formulas from the formula sheet. Little or no credit will be awarded for answers without the corresponding notation. 1. The following data shows the average annual employee income M ( t, v ) thousand dollars at a company where t is the number of years the employee has been with the company and v is the employee review value assigned to the employee. Employee Review Value (v) t years a. Find the quadratic cross-sectional model that could be used to model the boxed in data on the table above. Completely define your model by filling in all of the blanks below. (6 pts) M (, ) quadratic function with coefficients rounded to three decimal places gives the output units output description when input description and units and the employee s review value is 4,. input interval / domain b. Find the average annual income for an employee who has been with the company for 3.5 years and has an employee review value of 4. Show the mathematical notation, round your answer to one decimal place and include units. (3 pts) P a g e 10 13

7 2. Determine if f ( x ) is a valid probability density function. Show specific work (graph, values, etc.) to justify your conclusion. (4 pts) Is f ( x ) a valid p.d.f.? Justification: Yes or No 3. The amount of snowfall in feet is a remote region of Alaska in the month of January has the 2 2x 2x if 0 x 3 probability density function f ( x) 3 9. Check: f (1.5) otherwise a. Find the probability that this region receives more than 1.75 feet of snow in January. (4 pts) Show the specific probability and mathematical notation. Round your answer to four decimal places. b. Find the average amount of snow this region receives in January. (4 pts) Show the specific mathematical notation and include units with the answer. c. Find the variance in the amount of snow this region receives in January. (4 pts) Show the specific mathematical notation and give the answer. P a g e 11 13

8 4. R(s,n) million dollars describes the annual revenue for a fast food company, where s in the number of locations the company operates in South Carolina (SC) and n is the number of locations the company operates in North Carolina (NC). The partial derivatives for the annual revenue function are: Rs = 6s + 2n 100 million dollars per SC restaurant and Rn = 2s + 4n 80 million dollars per NC restaurant. The company currently operates 30 locations in SC and 20 locations in NC which generates an annual revenue of million dollars. a. Write the specific formula for dn. Your final answer should be in terms of n and s, not R. ds (4 pts) b. What are the units on dn? Circle one. ds (1 pt) NC restaurants SC restaurant SC restaurants NC restaurant million dollars NC restaurant million dollars SC restaurant c. Suppose the company is planning to close two locations in SC. In order to maintain the same revenue, how many new restaurants would they need to open in NC to compensate for the lost revenue in SC? Show all necessary work to solve this problem. (6 pts) Work: Conclusion: To maintain their annual revenue of million dollars, the company would have to open new locations in NC bringing the total number of restaurants to in NC and in SC. P a g e 12 13

9 5. The annual revenue generated at a college is given by 2 2 R x, y 1.5 x 2y 3xy 6x 18y million dollars when x thousand in-state students and y thousand out-of-state students are enrolled. Check: R(4,5) = 52 The first partial derivatives of R( x, y ) are R 3x 3y 6 and R 4y 3x 18. x a. Set up the system of equations that is used to find the critical point of R( x, y ). The equations should be in terms of x and y, not R. (2 pts) y b. Solve the system of equations to find the critical point of R( x, y ). Show all work (algebraic process or matrices). (5 pts) The critical point occurs when thousand in-state students and thousand out-of-state students are enrolled at the college. c. Find the second partial derivatives matrix and the value of the determinant at the critical point of R( x, y ). (7 pts) Matrix: Determinant: d. Classify the critical point by completing the statement below. (3 pts) The critical point identified in part b is a relative minimum or relative maximum or saddle point because and Reason 1: Show the specific value and comparison made (if necessary). Reason 2: Show the specific value and comparison made P a g e 13 13