Final Exam - LS108/1008
Chapters 1-10

	1)	Write the negation of the following sentence:
			She is not an Aquarius

		She is an Aquarius.

	2)	Construct the truth table for ~p~q

P	Q	NOT P	NOT Q	NOT P OR NOT Q	
0	0	1	1	1	
0	1	1	0	1	
1	0	0	1	1	
1	1	0	0	0	

	3)	Form the converse of the following:
			The car will not run if there is no gas.

			If the car will not run there is no gas

	4)	Using the symbolic representations:
			P: the fire will start
			Q: The wood is wet
		And the argument:
			The fire will start only if the wood is not wet
			The wood is wet.
			_____________________
			Therefore the fire will not start.
		Rewrite the argument in symbolic form.

	5)	Using the following sets:
			U={h,o,w,d,y}
			A={h,o,w}
			B={o,y}
		Find the following:
			a)  AB 
 {h, o, w, y}
			b)  A/
			{d,y}

	6)	When trying to pick a monitor, a printer, and a scanner for a new computer 		system, Pete narrowed his choices to 6 printers, 3 monitors, and 5 				scanners.  How many different ways could Pete set up his system?

		6x3x5=90 choices

	7)	A group of 9 girls and 2 boys must select a team of 4 people.  How many 			teams are possible if the team must consist of one boy and 3 girls?	

		2 Choose 1 x  9 Choose 3 = 2x84= 168

	8)	A jar contains 6 red marbles, 8 blue marbles, 11 white marbles and 9 			yellow marbles.  You pick a marble without looking:
			a)  What is the probability that it is white?
			Total = 34, White =11
			Probability of white = 11/34
 
			b)  Find the odds (not the same as probability) that it is not blue?
			Total =34, Not Blue = 26
			Probability of Not Blue = 26/34  = 13/17

9)	A card is drawn from a deck of cards (no jokers).  Find the probability that the card drawn is both above a 10 and red.
Above 10 = J,Q,K,A (of hearts and diamons) = 8 cards
Probability = 8/52 = 2/13

	10)	A pair of dice is rolled. Find the probability that the sum is 10 given that 			the sum is even.
Even sums = 2 (1 case), 4 (3 case), 6 (5 cases) ,8 (5 cases),10 (4 Cases),12 (1 Case)
Answer = 4/20 = 1/5

	11)	A random sampling of the speeds of 12 cars monitored on a small stretch 			of highway yielded the following data:
				40, 60, 50, 65, 60, 55, 60, 50, 70, 70, 55, 50
			a)  Find the mean speed.
			Total =685
			Count = 12
			Mean = 685/12 = 57.083

			b)  Find the median speed.
40	50	50	50	55	55	60	60	60	65	70	70	
			Middle Values 55 and 60: Average = 57.5
			c)  Construct a histogram of the data.

	12)	The cost per gallon of regular unleaded gasoline at area gas stations is 
			Price/gallon		Frequency
			   $1.50		   3
			   $1.60		   5
			   $1.65		   10
			   $1.68		   4
			   $1.70		   1
			a)  Find the standard deviation of the prices.	
	Price/gallon	Frequency	FX	FXX	
	$1.50 	3	$4.50 	6.75	
	$1.60 	5	$8.00 	12.8	
	$1.65 	10	$16.50 	27.225	
	$1.68 	4	$6.72 	11.2896	
	$1.70 	1	$1.70 	2.89	
Total	N/A	23	$37.42 	60.9546	
		Standard Deviation:
			b)  Find the mean price/gallon.
			Mean = 37.42/23 = 1.627

	13)	Refer to the standard normal z-distribution.  What percentage lies between 		z=0 and z=1.57?
		P(Z<=1.57) - .5
		.9418 -.5 
		.4418

	14)	400 people were surveyed and 310 said that they favor a referendum to 			raise taxes to pay for a new gymnasium.  Assuming a 90% level of 				confidence, find the margin of error, to the nearest tenth of a percent, 			associated with the sample proportion.

		Using the 90% CI we get 1.65
		E =  or 3.445%
		So interval is 

	15)	A deposit of $1100 is in a bank for 20 years earning interest at a rate of 			6.5%.  Find the interest earned if the interest is:
 
			I = PxRxT = 1100 x .065 x 20 = $1430

			b)  compounded monthly.
			20 years = 20x12 = 240 months
			Interest per month = .0054166
			Final Amount = 
			Interest = $3912.09 

	16)	Find the future value of an ordinary annuity with monthly payments of 			$200 that earns 8% interest and paid into for 20 years.
		Monthly Interest = .006667
		Months = 20x12= 240
		Future Value = 

	17)	If the investment considered in problem 16 is then withdrawn on a 				monthly basis at an amount of $300/month, how many years will the 			money last (assume the funds continue to earn the same 8% rate)?

	18)	Find the length of the longest object (assume this object to be one 				dimensional) that will fit inside a rectangular box that is 4 ft long 3 ft wide 		and 5 ft high.
		Will be of maximum dimension so 5ft

	19)	Solve the system:
			x+y=6
			x-3y=2
			Subtract Equation 2 from Equation 1
			4y=4 so y =1
			Use this value in equation 1 to get x=5
			Solution (5,1)

	20)	Given the system of linear inequalities:
				y≤-3x+6
				-y≥x-4
				x≥0
				y≥0
			a)  Graph the region of solutions
			
			b)  Find all of the region’s corner points
			Corner Points: (2,0), (1,3) and (0,4)
			c)  Maximize P = 6.7x+4.5y over the region.
			Try all points 
P(2,0) =13.4
P(1,3) = 20.2
P(0,4) = 18

Answer x=1, y=3

	21)	Given the following information:
				x+y≤8
				2x+3y≤20
				x≥0
				y≥0
			and z = 5x+4y
		Maximize z.

P(7,0) = 35
P(4,4) = 36
P(8,0) = 40
Solution x=8,y=0

	22)	A woodworker makes two types of carvings, ducks and hawks.  Carving 			one duck requires 4 hours and materials that cost $42.  Carving one hawk 			requires 5 hours  and materials that cost $48.  The woodworker wants to 			work no more than 60 hours/week and spend no more than $900 for 			materials.  If his profit for a duck is $60 and his profit for a hawk is $66,
			a)  express this information in the form of an objective function 				which can be maximized and the appropriate constraint 					inequalities.

			Let x be the number of duck carvings
			Let y be the number of hawk carvings
				4x + 5y ≤ 60
				42x + 48y ≤ 900
				x≥0
				y≥0
			Maximize: 60x + 66y
			
			b) maximize the objective function.
			
		Max profit is when 15 hawks are created: 15x66 = $990

	23)	Find the value of v where v = log7 (1/49).

	24)	Solve the equation 6*10-0.8x = 60.

	25)	A certain strain of bugs had a population of 120.  One month later the 			population had risen to 150.  Find the values of a and b in the model P = 			aebt that represents the population after t months.

		

	26)	A particular radioactive substance decayed from 100 grams to 20 grams in 		6 weeks.  Find the half-life of the substance.

		Half Life = (ln 2)/k = 2.58weeks

	27)	Given y = 2x2 +x -1:
			a)  Find the slope of the secant line through x = 1 and x = 3.
			At x=1, y=2
			At x=3, y=20
			Slope = 18/2 = 9
			b)  Find the equation for that secant line.
			Find intercept:
			2 = 9(1) +k
			K=-7
			So equation: y=9x-7

28)	A rock falls from a cliff 144 ft high. How long does it take the rock to 			reach the ground?  
Assuime gravity/acceleration is 9.8m/s-squared