Mathematics for the Trades: A Guided Approach (11th Edition) (What's New in Trade Math)
11th Edition
ISBN: 9780134756967
Author: Hal Saunders, Robert Carman
Publisher: PEARSON
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Chapter 3.2, Problem 16FE
Construction A 4-ft by 8-ft piece of
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Let G
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(V,E) be a 3-connected graph. We prove that for every x, y, z Є V, there is a
cycle in G on which x, y, and z all lie.
(a) First prove that there are two internally disjoint xy-paths Po and P₁.
(b) If z is on either Po or P₁, then combining Po and P₁ produces a cycle on which
x, y, and z all lie. So assume that z is not on Po and not on P₁. Now prove that
there are three paths Qo, Q1, and Q2 such that:
⚫each Qi starts at z;
• each Qi ends at a vertex w; that is on Po or on P₁, where wo, w₁, and w₂ are
distinct;
the paths Qo, Q1, Q2 are disjoint from each other (except at the start vertex
2) and are disjoint from the paths Po and P₁ (except at the end vertices wo,
W1, and w₂).
(c) Use paths Po, P₁, Qo, Q1, and Q2 to prove that there is a cycle on which x, y, and
z all lie. (To do this, notice that two of the w; must be on the same Pj.)
Chapter 3 Solutions
Mathematics for the Trades: A Guided Approach (11th Edition) (What's New in Trade Math)
Ch. 3.1 - Write as a decimal number: seven tenthsCh. 3.1 - You have three dollars and seventy-five cents and...Ch. 3.1 - Write in words. 0.72Ch. 3.1 - Write in words. 8.7Ch. 3.1 - Write in words. 12.36Ch. 3.1 - Prob. 4AECh. 3.1 - Write in words. 3.072Ch. 3.1 - Write in words. 14.091Ch. 3.1 - Write in words. 3.0024Ch. 3.1 - Write in words. 6.0083
Ch. 3.1 - Prob. 9AECh. 3.1 - Write as a decimal number. Three and four tenthsCh. 3.1 - Prob. 11AECh. 3.1 - Prob. 12AECh. 3.1 - Prob. 13AECh. 3.1 - Prob. 14AECh. 3.1 - Prob. 15AECh. 3.1 - Prob. 16AECh. 3.1 - Write as a decimal number. One hundred sixteen...Ch. 3.1 - Prob. 18AECh. 3.1 - Write as a decimal number. Two and three hundred...Ch. 3.1 - Write as a decimal number. Ten and two hundred...Ch. 3.1 - Add or subtract as shown. 14.21 + 6.8Ch. 3.1 - Prob. 2BECh. 3.1 - Add or subtract as shown. 2.83 + 12.19Ch. 3.1 - Add or subtract as shown. 52.37 + 98.74Ch. 3.1 - Add or subtract as shown. 0.687 + 0.93Ch. 3.1 - Prob. 6BECh. 3.1 - Add or subtract as shown. 507.18 + 321.42Ch. 3.1 - Prob. 8BECh. 3.1 - Add or subtract as shown. 45.6725 + 18.058Ch. 3.1 - Prob. 10BECh. 3.1 - Add or subtract as shown. 19 12.03Ch. 3.1 - Prob. 12BECh. 3.1 - Add or subtract as shown. 33.40 18.04Ch. 3.1 - Prob. 14BECh. 3.1 - Add or subtract as shown. 75.08 32.75Ch. 3.1 - Prob. 16BECh. 3.1 - Add or subtract as shown. 30 7.98Ch. 3.1 - Prob. 18BECh. 3.1 - Add or subtract as shown. 130 16.04Ch. 3.1 - Add or subtract as shown. 19 5.78Ch. 3.1 - Add or subtract as shown. 37 + 0.09 + 3.5 + 4.605Ch. 3.1 - Prob. 22BECh. 3.1 - Add or subtract as shown. 14.75 + 9 + 3.76Ch. 3.1 - Prob. 24BECh. 3.1 - Add or subtract as shown. 68.708 + 27.18Ch. 3.1 - Prob. 26BECh. 3.1 - Add or subtract as shown. 47.04 31.88Ch. 3.1 - Add or subtract as shown. 180.76 94.69Ch. 3.1 - Add or subtract as shown. 26.45 17.832Ch. 3.1 - Prob. 30BECh. 3.1 - Add or subtract as shown. 6.4 + 17.05 + 7.78Ch. 3.1 - Add or subtract as shown. 212.4 + 76 + 3.79Ch. 3.1 - Add or subtract as shown. 26.008 8.4Ch. 3.1 - Prob. 34BECh. 3.1 - Add or subtract as shown. 0.0046 + 0.073Ch. 3.1 - Add or subtract as shown. 0.038 + 0.00462Ch. 3.1 - Add or subtract as shown. 28.7 7.38 + 2.9Ch. 3.1 - Add or subtract as shown. 0.932 + 0.08 0.4Ch. 3.1 - Add or subtract as shown. 6.01 3.55 0.712Ch. 3.1 - Prob. 40BECh. 3.1 - Machine Trades What is the combined thickness of...Ch. 3.1 - Electrical Trades The combined weight of a spool...Ch. 3.1 - Electrical Trades The following are diameters of...Ch. 3.1 - Trades Management In estimating the cost of a job,...Ch. 3.1 - Metalworking Find A, B, and C.Ch. 3.1 - Plumbing A piece of pipe 8.4 in. long is cut from...Ch. 3.1 - Machine Trades A certain machine part is 2.345 in....Ch. 3.1 - Masonry Find the total cost of the materials for a...Ch. 3.1 - Masonry The specifications for a reinforced...Ch. 3.1 - Construction A plot plan of a building site showed...Ch. 3.1 - Automotive Trades A mechanic must estimate the...Ch. 3.1 - Automotive Trades A heated piston measures 8.586...Ch. 3.1 - Prob. 13CECh. 3.1 - Prob. 14CECh. 3.1 - Prob. 15CECh. 3.1 - Prob. 16CECh. 3.1 - Prob. 17CECh. 3.1 - Prob. 18CECh. 3.1 - Automotive Trades A technician renovating a truck...Ch. 3.1 - Prob. 20CECh. 3.1 - Automotive Trades An automotive technician is paid...Ch. 3.1 - Prob. 22CECh. 3.1 - Prob. 23CECh. 3.1 - Life Skills Balance this checkbook by finding the...Ch. 3.1 - Automotive Trades What is the actual cost of the...Ch. 3.1 - Add as shown. (a)...Ch. 3.2 - What is half of 0.8?Ch. 3.2 - Prob. 2LCCh. 3.2 - Prob. 3LCCh. 3.2 - Multiply or divide as shown. 0.01 0.001Ch. 3.2 - Multiply or divide as shown. 10 2.15Ch. 3.2 - Multiply or divide as shown. 0.04 100Ch. 3.2 - Prob. 4AECh. 3.2 - Multiply or divide as shown. 0.7 1.2Ch. 3.2 - Prob. 6AECh. 3.2 - Multiply or divide as shown. 0.003 0.01Ch. 3.2 - Multiply or divide as shown. 7.25 0.301Ch. 3.2 - Multiply or divide as shown. 24.6 8.5Ch. 3.2 - Prob. 10AECh. 3.2 - Multiply or divide as shown. 0.6 0.6 6.0Ch. 3.2 - Prob. 12AECh. 3.2 - Multiply or divide as shown. 3.618 0.6Ch. 3.2 - Prob. 14AECh. 3.2 - Multiply or divide as shown. 4.40 0.22Ch. 3.2 - Prob. 16AECh. 3.2 - Multiply or divide as shown. 0.0405 0.9Ch. 3.2 - Prob. 18AECh. 3.2 - Multiply or divide as shown. 4 0.01Ch. 3.2 - Multiply or divide as shown. 2.59 70Ch. 3.2 - Multiply or divide as shown. 44.22 6.7Ch. 3.2 - Multiply or divide as shown. 104.2 0.0320Ch. 3.2 - Multiply or divide as shown. 484 0.8Ch. 3.2 - Prob. 24AECh. 3.2 - Prob. 25AECh. 3.2 - Prob. 26AECh. 3.2 - Prob. 27AECh. 3.2 - Prob. 28AECh. 3.2 - Prob. 29AECh. 3.2 - Prob. 30AECh. 3.2 - Multiply or divide as shown. 0.0092 115Ch. 3.2 - Prob. 32AECh. 3.2 - Prob. 33AECh. 3.2 - Prob. 34AECh. 3.2 - Prob. 35AECh. 3.2 - Prob. 36AECh. 3.2 - Multiply or divide as shown. 0.062 27.5Ch. 3.2 - Calculate using the proper order of operations....Ch. 3.2 - Calculate using the proper order of operations....Ch. 3.2 - Calculate using the proper order of operations....Ch. 3.2 - Prob. 4BECh. 3.2 - Calculate using the proper order of operations....Ch. 3.2 - Calculate using the proper order of operations....Ch. 3.2 - Round the following numbers as indicated. 42.875...Ch. 3.2 - Round the following numbers as indicated. 0.5728...Ch. 3.2 - Round the following numbers as indicated. 6.54...Ch. 3.2 - Round the following numbers as indicated. 117.6252...Ch. 3.2 - Round the following numbers as indicated. 79.135...Ch. 3.2 - Round the following numbers as indicated. 1462.87...Ch. 3.2 - Round the following numbers as indicated. 3.64937...Ch. 3.2 - Round the following numbers as indicated. 19.4839...Ch. 3.2 - Round the following numbers as indicated. 0.2164...Ch. 3.2 - Prob. 10CECh. 3.2 - Divide and round as indicated. Round to two...Ch. 3.2 - Prob. 2DECh. 3.2 - Divide and round as indicated. Round to two...Ch. 3.2 - Prob. 4DECh. 3.2 - Divide and round as indicated. Round to two...Ch. 3.2 - Divide and round as indicated. Round to two...Ch. 3.2 - Divide and round as indicated. Round to two...Ch. 3.2 - Prob. 8DECh. 3.2 - Prob. 9DECh. 3.2 - Prob. 10DECh. 3.2 - Prob. 11DECh. 3.2 - Prob. 12DECh. 3.2 - D. Divide and round as indicated. Round to the...Ch. 3.2 - Prob. 14DECh. 3.2 - Prob. 15DECh. 3.2 - Prob. 16DECh. 3.2 - D. Divide and round as indicated. Round to three...Ch. 3.2 - Prob. 18DECh. 3.2 - Prob. 19DECh. 3.2 - Prob. 20DECh. 3.2 - Prob. 1EECh. 3.2 - Prob. 2EECh. 3.2 - E. Find the average for each of the following sets...Ch. 3.2 - Prob. 4EECh. 3.2 - E. Find the average for each of the following sets...Ch. 3.2 - Prob. 6EECh. 3.2 - Painting A gas-powered airless paint sprayer is...Ch. 3.2 - Allied Health How much total medication will a...Ch. 3.2 - Prob. 3FECh. 3.2 - Prob. 4FECh. 3.2 - Prob. 5FECh. 3.2 - Machine Trades How much does 15.7 sq ft of No. 16...Ch. 3.2 - Prob. 7FECh. 3.2 - Prob. 8FECh. 3.2 - Prob. 9FECh. 3.2 - Prob. 10FECh. 3.2 - Prob. 11FECh. 3.2 - Prob. 12FECh. 3.2 - Prob. 13FECh. 3.2 - Prob. 14FECh. 3.2 - Prob. 15FECh. 3.2 - Construction A 4-ft by 8-ft piece of 12-in. gypsum...Ch. 3.2 - Prob. 17FECh. 3.2 - Office Services The Happy Hacker electronics...Ch. 3.2 - Prob. 19FECh. 3.2 - Prob. 20FECh. 3.2 - Prob. 21FECh. 3.2 - Prob. 22FECh. 3.2 - Prob. 23FECh. 3.2 - Prob. 24FECh. 3.2 - Prob. 25FECh. 3.2 - Prob. 26FECh. 3.2 - Prob. 27FECh. 3.2 - Prob. 28FECh. 3.2 - Prob. 29FECh. 3.2 - Automotive Trades A 2017 Hyundai Sonata Eco is...Ch. 3.2 - Prob. 31FECh. 3.2 - Construction A certain thickness of sheet glass...Ch. 3.2 - Construction The Passive House Standard is a...Ch. 3.2 - Prob. 34FECh. 3.2 - Prob. 35FECh. 3.2 - Prob. 36FECh. 3.2 - Prob. 37FECh. 3.2 - HVAC A homeowner in the Northeast recently spent...Ch. 3.2 - Prob. 39FECh. 3.2 - Prob. 40FECh. 3.2 - Prob. 41FECh. 3.2 - Prob. 42FECh. 3.2 - Prob. 43FECh. 3.2 - Prob. 44FECh. 3.2 - Prob. 45FECh. 3.2 - Prob. 1GECh. 3.2 - Prob. 2GECh. 3.2 - Prob. 3GECh. 3.2 - Prob. 4GECh. 3.2 - Prob. 5GECh. 3.2 - Prob. 6GECh. 3.2 - Prob. 7GECh. 3.2 - Prob. 8GECh. 3.2 - Prob. 9GECh. 3.2 - Prob. 10GECh. 3.2 - Prob. 11GECh. 3.2 - Prob. 12GECh. 3.2 - Prob. 13GECh. 3.2 - Prob. 14GECh. 3.2 - Prob. 15GECh. 3.2 - Prob. 16GECh. 3.2 - Prob. 17GECh. 3.2 - Prob. 18GECh. 3.2 - Prob. 19GECh. 3.2 - Prob. 20GECh. 3.2 - Prob. 21GECh. 3.2 - Prob. 22GECh. 3.2 - Prob. 23GECh. 3.2 - Prob. 24GECh. 3.2 - Prob. 25GECh. 3.3 - Write 14 as a decimal number.Ch. 3.3 - What is one-third of 1.5?Ch. 3.3 - Write as decimal numbers. Use the bar notation for...Ch. 3.3 - Write as decimal numbers. Use the bar notation for...Ch. 3.3 - Write as decimal numbers. Use the bar notation for...Ch. 3.3 - Write as decimal numbers. Use the bar notation for...Ch. 3.3 - Write as decimal numbers. Use the bar notation for...Ch. 3.3 - Write as decimal numbers. Use the bar notation for...Ch. 3.3 - Write as decimal numbers. Use the bar notation for...Ch. 3.3 - Prob. 8AECh. 3.3 - Write as decimal numbers. Use the bar notation for...Ch. 3.3 - Write as decimal numbers. Use the bar notation for...Ch. 3.3 - Write as decimal numbers. Use the bar notation for...Ch. 3.3 - Prob. 12AECh. 3.3 - Write as decimal numbers. Use the bar notation for...Ch. 3.3 - Prob. 14AECh. 3.3 - Write as decimal numbers. Use the bar notation for...Ch. 3.3 - Prob. 16AECh. 3.3 - Prob. 17AECh. 3.3 - Write as decimal numbers. Use the bar notation for...Ch. 3.3 - Write as decimal numbers. Use the bar notation for...Ch. 3.3 - Prob. 20AECh. 3.3 - Write as decimal numbers. Use the bar notation for...Ch. 3.3 - Prob. 22AECh. 3.3 - Write as decimal numbers. Use the bar notation for...Ch. 3.3 - Prob. 24AECh. 3.3 - Prob. 25AECh. 3.3 - Prob. 26AECh. 3.3 - Write as decimal numbers. Use the bar notation for...Ch. 3.3 - Write as decimal numbers. Use the bar notation for...Ch. 3.3 - Prob. 1BECh. 3.3 - Calculate in decimal form. Round to agree with the...Ch. 3.3 - Calculate in decimal form. Round to agree with the...Ch. 3.3 - Prob. 4BECh. 3.3 - Calculate in decimal form. Round to agree with the...Ch. 3.3 - Calculate in decimal form. Round to agree with the...Ch. 3.3 - Calculate in decimal form. Round to agree with the...Ch. 3.3 - Prob. 8BECh. 3.3 - Calculate in decimal form. Round to agree with the...Ch. 3.3 - Prob. 10BECh. 3.3 - Calculate in decimal form. Round to agree with the...Ch. 3.3 - Prob. 12BECh. 3.3 - Allied Health If one tablet of calcium...Ch. 3.3 - Flooring and Carpeting Estimates of matched or...Ch. 3.3 - Roofing If a roofer lays 1012 squares of shingles...Ch. 3.3 - Machine Trades A bearing journal measures 1.996...Ch. 3.3 - Interior Design Complete the following invoice for...Ch. 3.3 - Construction A land developer purchased a piece of...Ch. 3.3 - Machine Trades Find the corner measurement A...Ch. 3.3 - Automotive Trades A cylinder is normally 3716 in....Ch. 3.3 - Plumbers deal with measurements in inches and...Ch. 3.3 - Plumbing To fit a 45 degree connection, a plumber...Ch. 3.3 - Electronics The frequency of a waveform or...Ch. 3.3 - Water/Wastewater Treatment A reservoir is used at...Ch. 3.3 - Sheet Metal Trades At one time the thickness of...Ch. 3.3 - Office Services Machine helpers at the CALMAC Tool...Ch. 3.3 - Prob. 15CECh. 3.3 - Carpentry For interior trim jobs, some carpenters...Ch. 3.3 - C. Practical Problems General Trades Use the table...Ch. 3 - (a) Write or pronounce a decimal number in words....Ch. 3 - Add, subtract, multiply, and divide decimal...Ch. 3 - Find averages. Find the average of 4.2, 4.8, 5.7,...Ch. 3 - Work with decimal fractions. (a) Write as a...Ch. 3 - Solve practical problems involving decimal...Ch. 3 - Write in words. 0.9Ch. 3 - Prob. 2APSCh. 3 - A. Write in words. 23.164Ch. 3 - A. Write in words. 63.219Ch. 3 - A. Write in words. 9.3Ch. 3 - A. Write in words. 3.45Ch. 3 - Write in words. 10.0625Ch. 3 - Write in words. 15.037Ch. 3 - Write as decimal numbers. Seven hundredthsCh. 3 - Write as decimal numbers. Eighteen thousandthsCh. 3 - Write as decimal numbers. Two hundred and eight...Ch. 3 - Write as decimal numbers. Sixteen and seventeen...Ch. 3 - Write as decimal numbers. Sixty-three and...Ch. 3 - Prob. 14APSCh. 3 - Write as decimal numbers. Five and sixty-three...Ch. 3 - Prob. 16APSCh. 3 - Prob. 1BPSCh. 3 - B. Add or subtract as shown. 18.8 + 156.16Ch. 3 - B. Add or subtract as shown. 7.52 + 11.77Ch. 3 - B. Add or subtract as shown. 26 + 0.06Ch. 3 - B. Add or subtract as shown. 3.68 1.74Ch. 3 - Add or subtract as shown. 12.46 8.51Ch. 3 - Prob. 7BPSCh. 3 - Prob. 8BPSCh. 3 - Prob. 9BPSCh. 3 - Add or subtract as shown. 0.232 + 5.079Ch. 3 - Prob. 11BPSCh. 3 - Prob. 12BPSCh. 3 - Prob. 13BPSCh. 3 - Prob. 14BPSCh. 3 - Prob. 15BPSCh. 3 - Prob. 16BPSCh. 3 - B. Add or subtract as shown. 315+1.08Ch. 3 - Prob. 18BPSCh. 3 - Prob. 19BPSCh. 3 - B. Add or subtract as shown. 8.1+0.47145Ch. 3 - Prob. 1CPSCh. 3 - C. Calculate as shown. 0.06 0.05Ch. 3 - Prob. 3CPSCh. 3 - Prob. 4CPSCh. 3 - Prob. 5CPSCh. 3 - C. Calculate as shown. 1.17 4.5Ch. 3 - Prob. 7CPSCh. 3 - Prob. 8CPSCh. 3 - Prob. 9CPSCh. 3 - C. Calculate as shown. 0.078 0.3Ch. 3 - Prob. 11CPSCh. 3 - Prob. 12CPSCh. 3 - Prob. 13CPSCh. 3 - Prob. 14CPSCh. 3 - 4.2+4.61.22.7321+0.41Ch. 3 - 7.23.251.33.5+5.74.02.8Ch. 3 - C. Calculate as shown. 1.2 + 0.7 2.2 + 1.6Ch. 3 - Prob. 18CPSCh. 3 - Prob. 19CPSCh. 3 - Prob. 20CPSCh. 3 - Round to two decimal digits. 0.007 0.03Ch. 3 - Prob. 22CPSCh. 3 - Round to two decimal digits. 17.8 6.4Ch. 3 - Prob. 24CPSCh. 3 - Round to three decimal places. 27.0072Ch. 3 - Prob. 26CPSCh. 3 - Round to three decimal places. 0.04 0.076Ch. 3 - Prob. 28CPSCh. 3 - Round to three decimal places. 17.6 0.082Ch. 3 - Prob. 30CPSCh. 3 - Round to the nearest tenth. 47.233Ch. 3 - Prob. 32CPSCh. 3 - Round to the nearest tenth. 0.08 0.053Ch. 3 - Prob. 34CPSCh. 3 - Round to the nearest tenth. 18.76 4.05Ch. 3 - Round to the nearest tenth. 0.91 0.97Ch. 3 - D. Write as a decimal number. 116Ch. 3 - Prob. 2DPSCh. 3 - D. Write as a decimal number. 532Ch. 3 - Prob. 4DPSCh. 3 - D. Write as a decimal number. 118Ch. 3 - Prob. 6DPSCh. 3 - Write as a decimal number. 21120Ch. 3 - Write as a decimal number. 116Ch. 3 - Prob. 9DPSCh. 3 - Prob. 10DPSCh. 3 - Write as a decimal number. 2516Ch. 3 - Prob. 12DPSCh. 3 - Prob. 13DPSCh. 3 - Prob. 14DPSCh. 3 - Write in decimal form, calculate, round to two...Ch. 3 - Prob. 16DPSCh. 3 - Prob. 17DPSCh. 3 - Prob. 1EPSCh. 3 - E. Find the average for each of the following sets...Ch. 3 - Prob. 3EPSCh. 3 - Prob. 4EPSCh. 3 - Machine Trades A 516-in. bolt weighs 0.43 lb. How...Ch. 3 - Metalworking Twelve equally spaced holes are to be...Ch. 3 - Plumbing A plumber finds the length of pipe needed...Ch. 3 - Prob. 4FPSCh. 3 - Prob. 5FPSCh. 3 - Prob. 6FPSCh. 3 - Prob. 7FPSCh. 3 - Prob. 8FPSCh. 3 - Prob. 9FPSCh. 3 - General Trades What is the closest fractional...Ch. 3 - Prob. 11FPSCh. 3 - Trades Management When the owners of the Better...Ch. 3 - Prob. 13FPSCh. 3 - Sheet Metal Trades To determine the average...Ch. 3 - Prob. 15FPSCh. 3 - Prob. 16FPSCh. 3 - Prob. 17FPSCh. 3 - General Interest In 2015, many states, counties,...Ch. 3 - Prob. 19FPSCh. 3 - Automotive Trades What is the outside diameter d...Ch. 3 - Prob. 21FPSCh. 3 - Prob. 22FPSCh. 3 - Prob. 23FPSCh. 3 - Printing A 614-page book was printed on paper...Ch. 3 - Allied Health The primary care physician has...Ch. 3 - Prob. 26FPSCh. 3 - Prob. 27FPSCh. 3 - Prob. 28FPSCh. 3 - Prob. 29FPSCh. 3 - Plumbing The inside diameter (I.D.) and outside...Ch. 3 - Prob. 31FPSCh. 3 - Office Services A shop technician earns 18.26 per...Ch. 3 - Prob. 33FPSCh. 3 - Prob. 34FPSCh. 3 - Prob. 35FPSCh. 3 - Prob. 36FPSCh. 3 - Prob. 37FPSCh. 3 - HVAC In March 2012, home heating oil sold for an...Ch. 3 - Prob. 39FPSCh. 3 - Prob. 40FPSCh. 3 - Prob. 41FPSCh. 3 - Prob. 42FPSCh. 3 - Prob. 43FPSCh. 3 - Prob. 44FPSCh. 3 - Prob. 45FPSCh. 3 - General Interest In January 2016, the state of...Ch. 3 - Life Skills A certain electric company charges...Ch. 3 - Plumbing A plumber replaced a showerhead that used...Ch. 3 - Life Skills During the severe droughts in...Ch. 3 - Automotive Trades Parts (a) and (b) of this...Ch. 3 - Agriculture The table below shows the number of...Ch. 3 - Construction Over the course of a building...Ch. 3 - Metalworking Find the total length of the piece of...
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Algebra
ISBN:9780998625713
Author:Lynn Marecek, MaryAnne Anthony-Smith
Publisher:OpenStax - Rice University
What is a Prism? | Types of Prism | Don't Memorise; Author: Don't Memorise;https://www.youtube.com/watch?v=wxDTrgnyu28;License: Standard YouTube License, CC-BY