Algebra Questions

2 Observational Studies 2 1 In the following descriptions of a study confounding is present Describe the explanatory and confounding variable in the study and how the confounding may invalidate the conclusions of the study Furthermore suggest how you would change the study to eliminate the effect of the confounding variable a A prospective study is conducted to study the relationship between incidence of lung cancer and level of alcohol drinking The drinking status of 5 000 subjects is determined and the health of the subjects is then followed for 10 years The results are given below Drinking Status Heavy drinker Light drinker Total Lung Cancer Yes No 50 30 80 Total 2 150 2 200 2 770 2 800 4 920 5 000 b A study was conducted to examine the possible relationship between coronary disease and obesity The study found that the proportion of obese persons having developed coronary disease was much higher than the proportion of nonobese persons A medical researcher states that the population of obese persons generally has higher incidences of hypertension and diabetes than the population of nonobese persons

Exercise 2 2 1 Consider the set PT of all Pythagorean triplets For a b c PT we assume a bc Then V a b c PT a b c 60 a b c 60 a b c 3 4 5 Hint Prove that for every a b c PT we have that 3 a b and that 5 a b c Conclude that 60 a b c Observe also that a b c 30 a b c 3 4 5 2 Prove that if a b c E PT is such that a b c form an arithmetic progression then a b c 3k 4k 5k for some k N 0 Conclude that 3 4 5 is the only PPT whose components form such a progression

3 Let A 6 1 3 1 and u 3 C 5 15 B3 c 23 a Compute Au and Ac if possible If not possible explain why not b Does the matrix equation Ar c have a solution c Does the matrix equation Ar b have a solution for any choice of b R d Restate the above question as a computationally equivalent question as done in page 2 of the notes

1 1 Exercises 1 The height above ground h of a bailt seconds after being dropped given by the equation h t 4 91 60 and shown on the graph below MO a How high is the ball at f 2 seconds b How long does it take the ball to hit the ground c When is the ball at a height of 10 meters d When will the height of the ball be greater than 10 meters e Using the graph estimate the height of the ball at t 1 second then use the equation fo above to find the exact value f Using the graph estimate at what time the ball will be a height of 20 meters then u formula for h above to find the exact value

Add or subtract Simplify by collecting like terms if possible 9 500 108 4 32 9 500 108 432 ce 50 4

6 6 17 X 30 18 yz 17 X 30 18 vyz View an example All parts showing 4 Simplify by taking roots of the numerator and the denominator Assume that all expressio 6 19 13 X 12 30 y z 6 12 30 yz Summarize 12 Finally simplify each radical if possible 6 x18 6x x 6 x X 19 25 y z 6 19 618 6 36

mplify the following expression 3 27x 27x 5 y Type an exact answer using radicals as needed 4 Simplify the following expression 3 11 1331x 3 3 NY Note that expression in the radical in the denominator is a perfect cube S root 3 133193 2 33 2 11x Y

Simplify the following expression 3 64x 3 5 64x 3 y 5 11 Type an exact answer using radicals as needed

The function f x x is one to one a Find an equation for f x the inverse function b Verify that the equation is correct by showing that f f x x and 1 f x x a Select the correct choice below and if necessary fill in the answer box to complete your choice A 1 x x for x 0 B 1 x for all x O c 1 x for x 0 b Verify that the equation is correct 1 1 x f x and X and The equation is not verified Part 3 of 3 verified 1 1 x 1 x X Substitute Simplify Y

Multiply and simplify Assume that no radicands were formed by raising negative numbers to even powers 4a b 1024a b Choose the correct expression below 4ab4b 4ab4a 4ab4b 4ab4a

3 3 5 5 8x 3 Type an exact answer using radicals as needed

What is the equation of the graph that represents the parent function f x x stretched vertically by a factor of 2 and then shifted up 3 spaces A g x 2 x 3 OB g x 2x 3 C g x 2x 3 OD g x 2 x 3 4

Next question Use the given equivalents along with dimensional analysis to convert the given unit to the unit indicated 18 oz to g 18 oz 16 oz 1 lb 2000 lb 1 T 1 oz 1 lb 1T 28 g 0 45 k 0 9 t

The approximate heights of two of the world s tallest mountains are given Determine which is the taller mountain and by how many meters Roull Manaslu 8163 meters and Gasherbrum 1 26 509 feet Click the icon to view a table of English and Metric Equivalents The taller mountain Round to the neares is taller by approximately Gasherbrum I Manaslu meters HOLD A English and Metric Equivalents English and Metric Equivalents 1 in 2 54 cm M X

Convert the given unit of weight to the unit indicated 16 g to cg 16 g cg

Convert the given unit of weight to the unit indicated 8 5 dg to mg 8 5 dg mg

14 F 5 rature to its equivalent temperature on the Celsius scale Use C F 32 or F C 32 where F is the degrees in Fahrenheit and C is the degrees in Celsi 14 F C Round to the nearest tenth as needed

mg The prescribed dosage of a drug is 12 daily meaning that 12 milligrams of the drug should be administered daily for each kilogram of a patient s weight How many 200 milligram tablets shou kg be given each day to a patient who weighs 164 pounds Use the fact that 1 lb 0 45 kg The patient should receive 200tabilligsarach day Round to the nearest whole number as needed

5 9 Convert the given Fahrenheit temperature to its equivalent temperature on the Celsius scale Use C F 32 or F C 32 where F is the degrees in Fahrenheit and C is the degrees in Celsius 43 F 43 F C nd to the nearest tenth as needed

Use the given equivalents along with dimensional analysis to convert the given unit to the unit indicated 45 lb to g 45 lb g 16 oz 1 lb 2000 lb 1T 1 oz 28 g 1 lb 0 45 kg 1T 0 91

se the 6 tbsp to mL Click to check the table of English and Metric Equivalents for Capacity 6 tbsp mL Round to two decimal places as needed English and Metric Equivalents for Capacity English and Metric Equivalents for Capacity 1 teaspoon tsp 5 milliliters mL 15 milliliters mL 30 milliliters mL 1 tablespoon tbsp 1 fluid ounce fl oz 1 cup c 1 pint pt 1 quart ot 0 24 liter L 0 47 liter L 0 95 liter L X

Lena 2735 miles Mekong 4328 kilometers Click the icon to view a table of English and Metric Equivalents Select the correct choice below and if necessary fill in the answer box to complete your choice OA Mekong is the longer river by approximately km Round to the nearest whole number as needed B Lena is the longer river by approximately km Round to the nearest whole number as needed OC Both rivers are the same length rounded the nearest km English and Metric Equivalents English and Metric Equivalents 1 in 2 54 cm 1 ft 30 48 cm

two of the world s wettest places are given Determine which location has the greater average rainfall and by how many inches Round to the nearest in Bowden Pen Jamaica 7 821 millimeters and Henderson Lake Canada 262 inches Click the icon to view a table of English and Metric Equivalents The average rainfall is greater in The average rainfall is greater by approximately Data table English and Metric Equivalents 1 in 1 ft 1 yd 1 mi 2 54 cm 30 48 cm 0 9 m 1 6 km inches X

Use the unit fractions 36 in 1 yd Convert 6 yd to cm and 2 54 cm 1 in 6 yd is cm Type an integer or decimal rounded to the nearest hundredth as needed

Express 80 kilometers per hour in miles per hour mi hr Round to the nearest hundredth as needed

Use dimensional analysis to convert the quantity to the indicated units 18 ft to in in Round to the nearest hundredth as needed

3 Graph the following function y 2X 20X 50 in the interval from X 0 to X 15

4 Graph the following inequality 20x 30y 420

Solve the proportion equation 2 ft 9 in 4 ft 7 in X cm 2 4 cm X

The following system has an infinite number of solutions Form an augmented matrix then write the matrix in the reduced form Write the reduced form of the matrix below and then write the solution in terms of z The required augmented matrix is where 11 1 a21 0 a315 and x 0 7 a12 2 1 4 a22 a32 2 3 3 a13 2 1 a23 1 a33 1x 2x 3x 3 2y 2z 4 4y 12 2 2y 3z 6 a11 a21 a31 1 1 a12 a13 b b2 a22 a23 a32 a33 b3 b b b3 4 1 2 6 y 2

Classify the number as one or more of the following natural number integer rational number or real number 7 9 Percent of students who graduate with honours from a university Of which of the following sets is 7 9 a member Select all that apply rational real natural integer

59 Angle Measure Find the measure in radians of a central angle of a sector of area 16 in in a circle of radius 3 0 in

solve the initial value problem y 4 3x 2 3y 2 12y y 0 2 and determine the interval in which the solution is valid

Concept Check Find the area of each sector 43 0 6 27

2 20 pts Rate of Change Look at the table below and answer the following questions Time yrs 1 Pop mil 3 7 Time yrs Pop mil 3 8 7 Mean 44 8 2 3 9 8 Increasing 3 6 Decreasing 3 4 1 9 independent 3 4 a Give the mean and median for each of the time frames 4 4 3 10 d What is the range for the dataset 3 2 Median 3 15 5 3 7 classes you are taking this semester what e Is there a trend we can gather from the data 11 3 8 b What would we represent to be the independent and dependent variables give units too 6 dependent 3 1 c From the table give when we increase and decrease use the format we used in class 12 4 2 is your

4 15 pts For some linear model we are given the following information The slope of the equation is 9 and it intersects at a point 5 3 Using our equation for solving slopes determine if the points exist on this same line a 3 8 b 3 15 c 7 21 d 4 6 asing 2 1 bns S F b 5 Extra Credit Out of the classes you are taking this semester what would you say is your favorite class and why is it mine just kidding but not really 20 bris 2

3 20 pts Solve for the slope of each of the data points If the slope doesn t exist write NA a 2 4 and 2 10 b 3 9 and 2 3 c 3 2 and 3 2 you see Include o might notice d 1 2 and 1 5 8A b y ai ysa uoy bluow farw eteemea airti grilst ens uoy seaaslo er to tuo libe10 six S ylisen ton Jud pribbix tau Tenim Ji al vriw bris easio ethovet

29 Rotating Wheels The rotation of the smaller wheel in the figure causes the larger wheel to rotate Through how many degrees does the larger wheel rotate if the smaller one rotates through 60 0 5 23 cm 8 16 cm

Write the following as a unit rate Express the answer as a fraction with a denominator of 1 Then express the answer in words using the word per 234 roses for every 9 vases Express the answer as a fraction Select the correct choice below and fill in the answer boxes to complete your choice Simplify your answers OA The unit rate is 26 rose s 1 vase s Express the answer in words using the word per B The unit rate is vase s rose s The unit rate is

Which of the following statements are correct Select all that apply 1 xb a xa b Oxab x x a 0 2 2 bx x b None of the Above

Use the graph of H shown to the right to find the limit When necessary state that the limit does not exist lim H X x 6 CLIED Select the correct choice below and if necessary fill in the answer box to complete your choice OA lim H x Type an Integer or a simplified fraction OB The limit does not exist

se the graph to answer these questions a Find lim f x and lim f x X 4 X 47 b Find lim f x X 4 c Find f 4 d Is f x continuous at x 4 Why or why not a Select the correct choice below and if necessary fill in the answer box to complete your choice OA lim f x X 4 OB The limit does not exist Select the correct choice below and if necessary fill in the answer box to complete your choice Simplify your answer OA lim X 4 OB The limit does not exist f x Simplify your answer b Select the correct choice below and if necessary fill in the answer box to complete your choice OA lim f x Simplify your answer X 4 OB The limit does not exist c Select the correct choice below and if necessary fill in the answer box to complete your choice OA 1 4 Simplify your answer OB The function is undefined at x 4 d Is f x continuous at x 4 Why or why not OA Yes f x is continuous at x 4 because f 4 exists OB Yes f x is continuous at x 4 because lim f x f 4 X 4 OC No f x is not continuous at x 4 because f 4 is undefined OD No f x is not continuous at x 4 because lim f x does not exist

Fill in the blanks The function gof x is found by replacing each occurrence of x in the equation for with

The volume of this triangular prism is 3 420 cubic meters What is the value of m 19 m m 20 m meters m

Consider the function f and its graph given to the right lim X 4 g x b lim X 4 If a limit does not exist state that fact Find a lim g x c g x X 4 a The notation lim g x is used to indicate the limit from the left that is where x a x a g x x 2 7X 2 2 6 2 12 for x 4 for x 24

Construct a consistent system of equations with a unique solution whose first equation is 0 5x1 0 3x2x3 0 8

Construct an inconsistent system of equations whose first equation is 0 5x1 0 3x2 x3 0 8 7

For the following matrix A 1 1 1 1 1 1 1 1 1 a Find the eigenvalues Explain how you can determine one eigen value A without any calculations b For each eigenvalue find the corresponding eigenvector s c Starting with the eigenvectors you found in b construct a set of orthonormal vectors using the Gram Schmidt procedure

olve the following system of linear equations using the substitution method 4x3 x 5x1 49 10x2 13x 30x3 10x 4x 5 4x

Compute the determinant of the following 3x3 matrix 4 1 51 8 8 7 10 7 5