Math 141 Week in ReviewSections 1.1-1.49/5/05Section 1.2.1. Find the slope of the line shown in the figure.Solution: Find two points on the line:(0, 6) and (3, 0)Use the slope formula:230061212xxyyxym2. Given the equation 3x - 2y = 7, answer the following questions:a. If x increases by 1 unit, what is the corresponding change in y?b. If x decreases by 2 units, what is the corresponding change in y?Solution: The relationship of the change in y to the change in x is about slope: a. 1?23m ? = 1.5 (increases by 1.5)b. 2?23m ? = -3 (decreases by 3)3. Find an equation of the vertical line that passes through (0, 8).Find an equation of the vertical line that passes through (2, 7).Solution: A vertical line has all x-values the same, so the equation is x=0 for the first one, and x=2 for the second one.4. Write the equation in slope-intercept form, and find the slope and y-intercept of the corresponding line: y + 6 = 0.Solution: Put the equation in slope-intercept form: y = 0x – 6. The slope is 0, and the y-intercept is (0, -6).5. Write the equation in slope-intercept form, and find the slope and y-intercept of the corresponding line: 8x + 5y – 24 = 0.Solution: 52458 xy slope: 58 y-intercept: 524,06. Find an equation of the line passing through the point (c, d) with undefined slope.Solution: A line with undefined slope is vertical. A vertical line has all x-values the same. So the equation is x = c.7. Find an equation of the line passing through the point (c, d) with slope 0.Solution: A line with slope 0 is horizontal. The equation is y = d.8. Sketch the straight line by finding the x- and y-intercepts: 3x – 5y = 20Solution: 9. A mathematical model for a pharmaceutical company’s sales, in billions of dollars, is given by S = 5.74 + 0.97x where x = 0 corresponds to 1988.a. What is the slope of the line? What does it represent?Solution: 0.97; On average, the increase in sales each year is $.97 billion.b. What is the S-intercept of the line? What does it represent?Solution: (0, 5.74); In 1988, the sales were $5.74 billion.10. The sales (in millions of dollars) of a company’s equipment sales from 2000 through 2004 is given below (x = 0 corresponds to 2000).Year x 0 1 2 3 4Annual Sales, y 2.8 4.1 5.3 6.2 7.6a. Plot the annual sales (y) versus the year (x).Solution:b. Draw a straight line L through the points corresponding to 2000 and 2004.c. Derive an equation of the line L.Solution: See graph for a) and b).2.1048.26.7my – 2.8 = 1.2(x – 0)y = 1.2x + 2.8d. Use the equation found in part (c) to estimate the annual sales of equipment in 2002.Solution: y = 1.2(2) + 2.8y = $5.2 millionSection 1.31. Determine whether the equation defines y as a linear function of x. If so, write it in the form y = mx + b. 3x = 2y - 7Solution: yes; y = (3/2)x + 7/22. Determine whether the equation defines y as a linear function of x.x - 5y = 2Solution: no3. An automobile purchased for use by the manager of a firm at a price of $26,000 is to be depreciated using the straight-line method over 5 yr. What will be the book value of the automobile at the end of 2 yr?Solution: 520005260000m y - 0 = -5200(x – 5) y = -5200x + 26000After 2 years: y = -5200(2) + 26000 = $15,6004. A camera manufacturer has a monthly fixed cost of $26,000 and a production cost of $12 for each camera manufactured. The cameras sell for $18 each.a. What is the cost function?Solution: C(x) = 26000 + 12xb. What is the revenue function?Solution: R(x) = 18xc. What is the profit function?Solution: P(x) = 18x – (26000 + 12x) = 6x - 26000d. Compute the profit (loss) corresponding to production levels of 2000, 6000, and 10,000 cameras, respectively.Solution: P(2000) = 6(2000) – 26000 = -14000P(6000) = 6(6000) – 26000 = 10000P(100000) = 6(10000) – 26000 = 340005. Sketch the equation of the demand curve 4p + 5x – 60 = 0, where x represents the quantity demanded in units of 1000 and p is the unit price in dollars. Determine the quantity demanded corresponding to the unit price $12.Solution: unitsxxxpxp24004.2154512154560546. The quantity demanded for a certain computer chip is 3000 units when the unit price is set at $20. The quantity demanded is 5200 units when the unit price is $13. Find the demand equation if it is known to be linear. Solution: (3000, 20) (5200, 13) 22007520030001320m 1132522007)3000(2200720xyxy7. Sketch the equation of the supply curve ½x – ¾p + 8 = 0, where x represents thequantity supplied in units of 1000 and p is the unit price in dollars. Determine thenumber of units of the commodity the supplier will make available in the market at the unit price $20.Solution: unitsxxxppx14000143323220332320843218. The manufacturer will make 2500 of the computer chips in problem #6 available when the price is $18. At a unit price of $15, 1800 chips will be marketed. Find the supply equation if the equation is known to be linear. How many chips will be marketed when the unit price is $22?Solution: (2500, 18) (1800, 15) 7003180025001518munitsxxxyxy34337527003227517003)2500(700318Section 1.41. Find the point of intersection of the pair of straight lines:2x + 3y = 12 5x – 2y = 11Solution: 2(2x + 3y = 12) 3(5x – 2y = 11) 4x + 6y = 2415x – 6y = 33 19x = 57 x = 3 2(3) + 3y = 12 y = 2(3, 2)2. Find the break-even point for the firm whose cost function C and revenue function R were found in Section 1.3, #4 above. Solution: 6x – 26000 = 0 x 4333R(x) = 18x = 18(26000/6) = $78,000(4333, $78000)3. A company manufactures microwave ovens. Each oven sells for $60. The monthly fixed costs total $24,000, and the variable cost of producing each oven is $8. Find the break-even point for the company.Solution: C(x) = 24000 + 8x R(x) = 60x24000 + 8x = 60xx 46260(462) = $27,692.31(462, $27,692.31)4. The sales for Maddie’s Beauty Supply are expected to be given by S = 3.2 + .04t thousand dollars t years from now. The annual sales of Jean’s Beauty Supply are expected to be given by S = 1.4 + .05t thousand dollars t years from now. When will Jean’s annual sales first surpass Maddie’s annual sales?Solution: 1.4 + .05t > 3.2 + .04tt > 180 years5.
View Full Document
Unlocking...