Find an equation in slope-intercept form (where possible) for each line in Exercises *Through (-8,1), with undefined slope

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1 *Must show all work to earn credit!* Find an equation in slope-intercept form (where possible) for each line in Exercises *Through (-8,1), with undefined slope A line with an undefined slope is parallel to the y-axis. The x-coordinate of all points on the line is the same. So, with the point (-8,1) on the line, the line must be: x = -8 *Use slopes to show that the square with vertices at (-2,5), (4,5), (4,-1), (-2, -1) has diagonals that are perpendicular. The diagonals run from (-2, -1) to (4, 5), and (-2, 5) to (4, -1) Slope of first line: [5 (-1)] / [4 (-2)] = 6/6 = 1 Slope of second line: (-1 5) / [4 (-2)] = -6/6 = -1 Since the two slopes are negative reciprocals of each other, the lines are perpendicular. *Graph each equation. 2x-3y = 12 To graph this by hand, rearrange the equation to isolate y on the left side: y = (2/3)x 4 Then pick a few points for x and calculate the y-coordinate: x = 0 y = -4 x = 3 y = -2 x = -3 y = -6 Plot the points (0, -4) and (3, -2) and connect with a line. The third point serves as a check. It should lie on the line as well.

2 The graph looks like this: *To achieve the maximum benefit for the heart when exercising, your heart rate (in beats per minute) should be in the target heart rate zone. The lower limit of this zone is found by taking 70% of the difference between 220 and your age. The upper limit is found by using 85%. a.find formulas for the upper and lower limits (u and l) as linear equations involving the age x. Let a = age in years, u = upper limit, l = lower limit u = 0.85(220 a) l = 0.70(220 a) b. What is the target heart rate zone for a 20-year-old? u = 0.85(220 a) = 0.85(220 20) = 170 l = 0.70(220 a) = 0.70(220 20) = 140 Target heart rate zone is 140 to 170. c. What is the target heart rate zone for a 40-year-old?

3 u = 0.85(220 a) = 0.85(220 40) = 153 l = 0.70(220 a) = 0.70(220 40) = 126 Target heart rate zone is 126 to 153. d. Two women in an aerobics class stop to take their pulse, and are surprised to find that they have the same pulse. One woman is 36 years older than the other and is working at the upper limit of her target heart rate zone. The younger woman is working at the lower limit of her target heart rate zone. What are the ages of the two women, and what is their pulse? Let a1 equal the age of the older woman Let a2 equal the age of the younger woman a1 a2 = 36 a1 = a (220 (a2 + 36)) = 0.70(220 a2) 0.85(184 a2) = 0.7(220 a2) (a2) = (a2) ( )(a2) = (a2) = 2.4 a2 = 16 a1 = = 52 The ages of the two women are 16 and 52, and their pulse is e. Run for 10 minutes, take your pulse, and see if it is in your target heart rate zone. (After all, this is listed as an exercise!) Why yes, yes it was. *Some scientists believe there is a limit to how long humans can live.* One supporting argument is that during the last century, life expectancy from age 65 has increased more slowly than life expectancy from birth, so eventually these two will be equal, at which

4 point, according to these scientists, life expectancy should increase no further. In 1900, life expectancy at birth was 46 yr, and life expectancy at age 65 was 76. In 2000, these figures had risen to 76.9 and 82.9, respectively. In both cases, the increase in life expectancy has been linear. Using these assumptions and the data given, find the maximum life expectancy for humans. Let L1 = life expectancy estimated at birth Let L2 = life expectance estimate at age 65 Y = number of years since 1900 L1 = 46 + ( )Y/100 = Y L2 = 76 + ( )Y/100 = Y The maximum life expectancy is where the two lines have the same value: L1 = L Y = Y ( )Y = Y = 30/0.24 = 125 The life expectancy at that point is: L1 = (125) = years L2 = (125) = years. Maximum life expectancy = years *Write a linear cost function for each situation. Identify all variables used. A parking garage charges 50 cents plus 35 cents per half-hour. Let C equal the total cost, and let t = the time parked in half-hours C(t) = 0.35t *Assume that each situation can be expressed as a linear cost function. Find the cost function in each case. Marginal cost: $90; 150 items cost $16,000 to produce. Marginal cost is the cost of each item produced. This is equal to the slope of the line. Using the cost for 150 items, we can find the intercept, b. C = total cost of production

5 x = number of items produced y = 90x + b 16,000 = 90(150) + b b = (150) b = The cost function is C = 90x *Let the supply and demand functions for butter pecan ice cream be given by p = s(q) = 2/5q and p = D(q) = 100 2/5q, where p is the price in dollars and q is the number of 10-gallon tubs. a. Graph these on the same axes. b. Find the equilibrium quantity and the equilibrium price. The equilibrium quantity is gallon tubs, at a price of 50 dollars per tub. *The manager of a restaurant found that the cost to produce 100 cups of coffee is $11.02, while the cost to produce 400 cups is $ Assume the cost C (x) is a linear function of x, the number of cups produced.

6 a. Find a formula for C (x). Slope of line = ( )/300 = C(x) = 0.097x + b Using the information given: C(100) = 0.097(100) + b = b = (0.097)(100) = 1.32 C(x) = 0.097x b. What is the fixed cost? The fixed cost is $1.32 c. Find the total cost of producing 1000 cups. C(1000) = 0.097(1000) = $98.32 d. Find the total cost of producing 1001 cups. C(1001) = 0.097(1001) = $98.42 e. Find the marginal cost of the 1001st cup. Marginal cost = C(1001) C(1000) = $ $98.32 = $0.10 f. What is the marginal cost of any cup and what does this mean to the manager? The marginal cost of any cup is $0.097 $0.10 This tells the manager that he has to sell each cup of coffee for at least this amount in order to make a profit. Selling cups at this price would only allow him to break even. * To produce x units of a religious medal costs C (x) = 12x The revenue is R(x) = 25x. Both C (x) and R (x) are in dollars. a. Find the break-even quantity. At the break-even point, cost equals revenue:

7 R(x) = C(x) 25x = 12x x = 39 x = 3 The break-even quantity is 3. b. Find the profit from 250 units. Profit = Revenue Cost P(x) = 25x (12x + 39) = 13x 39 P(250) = 13(250) 39 = $3,211 c. Find the number of units that must be produced for a profit of $130. P(x) = 13x 39 = x = x = 169 / 13 = 13 units * You may have heard that the average temperature of the human body is Recent experiments show that the actual figure is closer to The figure of 98.6 comes from experiments done by Carl Wunderlich in But Wunderlich measured the temperatures in degrees Celsius and rounded the average to the nearest degree, giving 37 c as the average temperature. a. What is the Fahrenheit equivalent of 37 c? ºF = (9/5)(ºC) + 32 = (9/5)(37) + 32 = 98.6 ºF b. Given that Wunderlich rounded to the nearest degree Celsius, his experiments tell us that the actual average human body temperature is somewhere between 36.5 c and 37.5 c. Find what this range corresponds to in degrees Fahrenheit. (9/5)(36.5) + 32 = 97.7 ºF

8 (9/5)(37.5) + 32 = 99.5 ºF Range = 97.7 to 99.5 ºF *The number of banks in the United States has dropped about 30% since The following data are from a survey in which x represents the years since 1900 and y corresponded to the number of banks, in thousands, in the United States. N = 10 X = 965 Y = 95.3 X ^ 2 = 93,205 XY = Y^2 = a. Find an equation of the least squares line. The equation of the least squares line has the form y = a + bx with: a = [ ( y)( x²) - ( x)( xy) ] / [ n( x²) - ( x)² ] a = [ (95.3)(93,205) - (965)(9165.1) ] / [ 10(93,205) - (965)² ] a = [ 38,115 ] / [ 825 ] a = 46.2 b = [ n( xy) - ( x)( y) ] / [ n( x²) - ( x)² ] b = [ 10(9165.1) - (965)(95.3) ] / [ 10(93,205) - (965)² ] b = [ ] / [ 825 ] b = The equation of the line is: y = x b. If the trend continues, how many banks will there be in 2004? For the year 2004, x = = 104 y = 46.2 (0.38)(104)

9 y = 6.68 Round this off to seven banks. c. Find and interpret the coefficient of correlation. The coefficient of correlation is calculated by: r = [ n( xy) - ( x)( y) ] / [ [ n( x²) - ( x)² ][ n( y²) - ( y)² ] ] r = [ 10(9165.1) - (965)(95.3) ] / [ [ 10(93205) - (965)² ][ 10(920.47) - (95.3)² ] ] r = [ / [ [ 825 ][ ] ] r = The coefficient of correlation ranges between -1 and +1. When the coefficient is close to +1, there is a strong positive relationship between the two variables. When the coefficient is close to -1, there is a strong negative relationship between the variables. This means that as one increases, the other decreases. The value of the coefficient in this problem is very close to -1, showing a strong negative relationship between the number of banks in the U.S. and the number of years since *While shopping for an air conditioner, Adam Bryer consulted the following table giving a machine s BTUs and the square footage (ft^2) that it would cool. Ft^2(x) BTUs ( y) a. Find the equation for the least squares line for the data. Using the data above, the following totals were calculated:

10 n = 10 x = 2,970 y = 72,500 x 2 = 974,650 y 2 = 546,250,000 xy = 22,912,500 Using the formulas for a and b shown in the previous problem: a = [ (72,500)(974,650) - (2,970)(22,912,500) ] / [ 10(974,650) - (2,970)² ] a = 2, b = [ 10(22,912,500) - (2,970)(72,500) ] / [ 10(974,650) - (2,970)² ] b = y = x b. To check the fit of the data to the line, use the results from part a to find the BTUs required to cool a room of 150 ft^2, 280 ft^2, and 420 ft^2. How well does the actual data agree with the predicted values? 150 ft 2 = ft 2 = ft 2 = The actual data agrees very well with the predicted values. Percent of error is 1.16% for 150 ft 2 and lower for the other two values. c. Suppose Adam s room measures 230 ft^2. Use the results from part a to decide how many BTUs it requires. If air conditioners are available only with the BTU choices in the table, which would Adam choose? y = (230) = 6, BTU Adam should choose the 6500 BTU unit, as the 6000 BTU unit will not meet his cooling needs. d. Why do you think the table gives ft^2 instead of ft^3, which would give the volume of the room?

11 Most rooms are a standard 8 feet high from floor to ceiling, so the requirements can be converted to a square footage basis. Most people know the size of their house in square feet, but would have no idea about what the actual volume is.