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Finite Math

(See attached file for full problem description)

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1. What is marginal cost? Fixed cost?

Find the slope for each line that has a slope.

3. Through (-2, 5) and (4, 7)
5. Through the origin (11, -2)
7. 2x + 3y = 15
9. y + 4 = 9
11. y = -3x

Find an equation in the form y = mx + b (where possible) for each line.

13. Through (5, -1); slope = 2/3
15. Through (5, -2) and (1, 3)
17. Through (-1, 4); undefined slope
19. Through (2, -1), parallel to 3x - y = 1
21. Through (2, -10), perpendicular to a line with undefined slope
23. Through (-7, 4), perpendicular to y = 8

Graph each linear equation as follows.

25. y = 6 - 2x
27. 2x + 7y = 14
29. y = 1
31. x +3y = 0

33. The supply and demand for crab meat in a local fish store are related by equations

Supply: p = S (q) = 6q + 3

And

Demand: p = D(q) = 19 - 2q,

Where p represents the price in dollars per pound and q represents the quantity of crabmeat in pounds per day. Find the supply and demand at each of the following prices:

a. $10
b. $15
c. $18
d. Graph both the supply and demand functions on the same axes.
e. Find the equilibrium price.
f. Find the equilibrium quantity.

35. For a new diet pill, 60 pills will be supplied at a price of $40, while 100 pills will be supplied at a price of $60. The demand for the diet pills is 50 pills at a price of $47.50 and 80 pills at a price of $32.50. Determine a linear demand function for these pills.

Find a linear cost function in the exercises below.

37. Eight units cost $300; fixed cost is $60.
39. Twelve units cost $445; 50 units cost $1585.
41. The cost of producing x cartons of CDs is C(x) dollars, where C(x) = 200x + 1000. The CDs sell for $400 per carton.
a. Find the break-even quantity.
b. What revenue will the company receive if it sells just that number of cartons?

43. The U.S. is China's largest export market. Imports from China have grown from about 19 billion dollars in 1991 to 102 billion dollars in 2001. This growth has been approximately linear. Use the given data pairs to write a linear equation that describes this growth in imports over the years. Let x = 91 represent 1991 and x = 101 represent 2001.
45. The U.S. Census Bureau reported that the median income for all U.S. households in 2000 was $42, 148. In 1993, the median income (in 2000 dollars) was $36, 746. The median income is approximately linear and is a function of time. Find a formula for the median income, I, as a function of the year x, where x is the number of years since 1900.
47. In general, people tend to live longer in countries that have a greater supply of food. Listed below is the 1997 daily calorie supply and 2000 life expectancy at birth for 10 randomly selected countries.

Country Calories (x) Life expectancy (y)

Afghanistan 1523 43
Belize 2862 74
Cambodia 1974 56
France 3551 79
India 2415 64
Mexico 3137 73
New Zealand 3405 78
Peru 2310 70
Sweden 3160 80
U.S. 3642 78

a. Find the coefficient of correlation. Do the data seem to fit a straight line?
b. Draw a scatterplot of the data. Combining this with your results from part a, do the data seem to fit a straight line?
c. Find the equation for the least squares line.
d. Use your answer from part c to predict the life expectancy in the United Kingdom, which has a daily calorie supply of 3237. Compare your answer with the actual value of 78 years.
e. Briefly explain why countries with a higher daily calorie supply might tend to have a longer life expectancy.
f. Find the coefficient of correlation and least squares line using data for a larger sample of countries, as found in an almanac or other reference. Is the result in general agreement with the previous results?

51. In general, the larger a state's population, the more its governor earns. Listed below are the estimated 2001 populations (in millions) and the salary of the governor (in thousands of dollars) for 8 randomly selected states.

a. Find the coefficient of correlation. Do the data seem to fit a straight line?
b. Draw a scatterplot of the data. Compare this with your answer from part a.
c. Find the equation for the least squares line.
d. Based on your answer to part c, how much does a governor's salary increase, on average, for each additional million in population?
e. Use your answer from part c to predict the governor's salary in your state. Based on your answers from parts a and b, would this prediction be very accurate? Compare with the actual salary, as listed in the almanac or other reference.
f. Find the coefficient of correlation and least squares line using data for all 50 states, as found in an almanac or other reference. Is the resulting general agreement with the previous results?
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(See attached file for full problem description)

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1. The marginal cost of an additional unit of output is the cost of the additional inputs needed to produce that output. The fixed cost is a cost that remains constant, regardless of any change in a company's activity.

3. The slope is (7-5)/(4-(-2))=2/6=1/3
4. The slope is 11/(-2)=-5.5
7. Since 2x+3y=15 implies that y=-(2/3)x+5, so the slope is -2/3
9. The slope is 0
11. The slope is -3

13. The equation of the line is y-(-1)=(2/3)(x-5), then we get y=(2/3)x-13/3
15. The slope is (3-(-2))/(1-4)=5/3, the the equation of the line is
y-3=(5/3)(x-1), then we get y=(5/3)x+4/3
17. The undefined slope means that the line has the form x=a. Since it goes through (-1,4), then the equation of the line is ...

Solution Summary

This solution is comprised of a detailed explanation to answer what is marginal cost and fixed cost.

$2.19