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The cost of hosting a dinner in a particular restaurant is given by y = 18.5x + 250, where x is the number of people at the dinner and y is dollars. What is the slope of this function? What does it mean in the context of the problem?Answer: Slope is 18.5, if the number of people increases by one the cost increases $18.5 dollars3. Write the equation of the line that is parallel to the x-axis and goes through the point (1,4).Answer: y = 44. Does the point (2,6) lie on the line that connects (1,4) and (0,3)? Explain.Answer: The equation of the line that connects the points is: y = x+3Then (2,6) does not lie in the line because 6 2+35. Which of the following pairs of lines are perpendicular? How do you know? a. y = 2/3x + 2 and y = -3x + 1b. y = 1/3x and y = -3x + 1 c. y = 3x and y = 1/3xAnswer:b) y = 1/3x and y = -3x + 1 are perpendicular because the product of their slopes is -1 (slopes are 1/3 and -3)6. Jeremy uses the linear function G = 12h + 50 to represent the grade, G (in points out of 100), that he can earn on an exam as a function of h, the number of hours he spends studying for the exam. Identify the slope and y-intercept of Jeremy's function and explain what they mean in the context of the problem.Slope is 12 , y-intercept is 50 If we increase by one hour the amount of time spent studying the grades increases by 12, and if we do not study the grade will be 50.b. If Jeremy spends 3 hours studying for the exam, what grade does he expect toearn? Show your work.If x = 3 then y = 12(3)+50 = 36+50 = 86Answer: 86c. How many hours should Jeremy study if he wants to earn a perfect score on the exam? Show your work.If y = 100 then 100 = 12h+50 then 12h = 50 then h = 25/6Answer: 25/6 hours7. Suppose that demand, D, for a particular product is given by the function D = 100 - 2p, where p is the price in dollars of the product and D is the number of products that can be sold at that price. a. What does the slope of this function mean in the context of the problem? The slope is -2, if the price increases by $1 dollar the demand decreases 2 unitsWhat price should be set in order to sell 75 items? Show your work. If D = 75 then 75 = 100-2p , then -25 = -2p , p = 12.5 Answer p = $12.58. Temperature may be given in degrees Celsius or in degrees Fahrenheit. The freezing point of water is 32°F or 0°C. The boiling point of water is 212°F or 100°C. Write the equation of a line that shows the relationship between degrees Fahrenheit and degrees Celsius. Answer: C = (5/9)(F-32)What is the temperature in Fahrenheit when it is 17?C?Answer: 62.6What is the temperature in Celsius when it is 79?F?Answer: 26.119. Write at least three different expressions that mean "slope." Answer: Gradient, inclination , obliquity10. In order to write the equation of a line, what two pieces of information do you need? (Hint: there may be more than one answer to this question.)Answer: Slope and one point11. The cost of manufacturing soccer balls is given by C = 24,000 + 7x, where x is the number of soccer balls produced. a. What is the slope of this equation and what does it represent in the context of the problem? Answer: Slope is 7 , if the number of balls increase 1 unit cost increases $7b. What is the y-intercept of this equation and what does it represent in the context of the problem?Answer: y-intercept is $2400 the cost is $2400 if we do not produce ballsc. If a manufacturer wanted to spend less than $30,000 to produce soccer balls, what is the maximum number of balls that can be produced? Show your work.If we want C < 30000 then 24,000 + 7x < 30,0007x < 6,000X < 857.14Answer: we can produce a maximum of 857 balls12. Write the equation of a line that is parallel to the line connecting (2,5) and (-1,-4). Answer: y = 3x-1 13. Describe what a sequence is, and give two example to support your definition.A sequence is a function where the domain are the natural numbersExamples: an = n2+1 , an = 3n+414. If you think of a sequence as a function, what is the domain?Answer: natural numbers15. What is a?5 in the following sequence?1, 4, 8, 13, 19, ... Answer: 26 (a0 = 1)16. Is the above sequence an arithmetic sequence? Why or why not?Answer: No, because the difference between consecutive terms is not constant17. Given the sequence an = n/(n + 1), what is a100?Answer: 100/10118. Give a recursive formula for the sequence 1, 4, 7, 10, 13, 16, ...Answer: a0 = 1 an = an-1 +3 for n ≥ 119.Find the first five terms of the following sequence:a?n = 2an-1 + 3a?1 = 2Answer: 2,7,17,37,7720. Find the sum that is given by: sigma (5 on top and k = 1 on the bottom) 3k – 1.Answer: 4021. Write the series 2 + 4 + 6 + 8 + ... + 100 using sigma notation.Answer: k=1k=502k22. The 50th term of an arithmetic sequence is 86, and the common difference is 2. Find the first three terms of the sequence.Answer: -48,-46 and -4423. If you save one penny on the first day of January, two pennies on the second day of January, three pennies on the third day of January, and continue with this pattern for a whole year, how much will you have saved by December 31?Answer: 2365 - 124. An arithmetic sequence has a ?1 = 5, and a?5 = 13. How many terms of this sequence must be added to obtain 60?Answer: 625. Suppose you purchase a car for $24,500, and it depreciates at the rate of $1,500 a year. Express the car's value after five years as a term in an arithmetic sequence.Answer: an = 24,500 -1500n , a5 = 24,500-1500(5) = 17,000 26. For the sequence 15, 22, 29, 36, 43, .. solve the following problems:a. Write a recursive formula for the sequence.b. Write an explicit formula for the sequence.c. Find the sum of the first 100 terms of the sequence.Answers:a) a1 = 15 , an = an-1 + 7 for n ≥ 2b) an = 8+7n for n ≥ 1c) 3615027. The fifth term of an arithmetic sequence is 100, and the common difference is 4. Find the first three terms of the sequence.Answer: 86, 88, 9228. Your starting annual salary for a new job is $40,000, and you are promised an annual raise of $1,500. What will your salary be at the beginning of your 11th year?Answer: $55,00029. Suppose your grandparents started a savings account for you on your first birthday with a deposit of $100. On your second birthday, they added $200, and on your third they added $300. They continued this pattern for your for your first 10 birthdays. Ignoring interest, how much is in your savings account after they make the deposit on your 10th birthday?Answer: $550030. Find common difference, fifth term, & sum of first 100 terms of the sequence:1, 2.5, 4, 5.5, ...Answers:Common difference = 1.5Fifth term = 7Sum of first 100 terms = 7525 ................
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