Elementary Technical Mathematics
11th Edition
ISBN: 9781285199191
Author: Dale Ewen, C. Robert Nelson
Publisher: Cengage Learning
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Textbook Question
Chapter 7, Problem 11T
A small gear with 36 teeth turns a large gear with 48 teeth at 150 rpm. What is the speed (in rpm) of the small gear?
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1. Abel's Theorem. The goal in this problem is to prove Abel's theorem by following a series of steps
(each step must be justified).
Theorem 0.1 (Abel's Theorem).
If y1 and y2 are solutions of the differential equation
y" + p(t) y′ + q(t) y = 0,
where p and q are continuous on an open interval, then the Wronskian is given by
W (¥1, v2)(t) = c exp(− [p(t) dt),
where C is a constant that does not depend on t. Moreover, either W (y1, y2)(t) = 0 for every t in I or
W (y1, y2)(t) = 0 for every t in I.
1. (a) From the two equations (which follow from the hypotheses),
show that
y" + p(t) y₁ + q(t) y₁ = 0 and y½ + p(t) y2 + q(t) y2 = 0,
2. (b) Observe that
Hence, conclude that
(YY2 - Y1 y2) + P(t) (y₁ Y2 - Y1 Y2) = 0.
W'(y1, y2)(t) = yY2 - Y1 y2-
W' + p(t) W = 0.
3. (c) Use the result from the previous step to complete the proof of the theorem.
Chapter 7 Solutions
Elementary Technical Mathematics
Ch. 7.1 - Express each ratio in lowest terms: 3 to 15Ch. 7.1 - Express each ratio in lowest terms: 6:12Ch. 7.1 - Express each ratio in lowest terms: 7:21Ch. 7.1 - Express each ratio in lowest terms: 422Ch. 7.1 - Prob. 5ECh. 7.1 - Prob. 6ECh. 7.1 - Prob. 7ECh. 7.1 - Express each ratio in lowest terms: 3 ft to 15 in.Ch. 7.1 - Express each ratio in lowest terms: 3 cm to 15 mmCh. 7.1 - Express each ratio in lowest terms: 1 in. to 8 ft
Ch. 7.1 - Prob. 11ECh. 7.1 - Prob. 12ECh. 7.1 - Prob. 13ECh. 7.1 - Prob. 14ECh. 7.1 - Prob. 15ECh. 7.1 - Express each ratio in lowest terms: 6 to 423Ch. 7.1 - Express each ratio in lowest terms: 513223Ch. 7.1 - Express each ratio in lowest terms: 1812214Ch. 7.1 - Express each ratio in lowest terms: 10 to 212Ch. 7.1 - Express each ratio in lowest terms: 78:916Ch. 7.1 - Express each ratio in lowest terms: 312 to 212Ch. 7.1 - Express each ratio in lowest terms: 223:334Ch. 7.1 - Prob. 23ECh. 7.1 - Prob. 24ECh. 7.1 - Express each rate in lowest terms: 240mi8galCh. 7.1 - Express each rate in lowest terms: 360mi18acresCh. 7.1 - Prob. 27ECh. 7.1 - Prob. 28ECh. 7.1 - Express each rate in lowest terms: 625mi1212hCh. 7.1 - Prob. 30ECh. 7.1 - Express each rate in lowest terms: 214lb6galCh. 7.1 - Prob. 32ECh. 7.1 - A bearing bronze mix includes 96 lb of copper and...Ch. 7.1 - What is the alternator-to-engine drive ratio if...Ch. 7.1 - A technician measured an oil flow rate of 90 gal...Ch. 7.1 - A flywheel has 72 teeth, and a starter drive-gear...Ch. 7.1 - Prob. 37ECh. 7.1 - Prob. 38ECh. 7.1 - Prob. 39ECh. 7.1 - The resistance in ohms of a resistor is the ratio...Ch. 7.1 - A 150-bu wagon holds 2.7 tons of grain. Express...Ch. 7.1 - The total yield from a 55-acre field is 7425 bu....Ch. 7.1 - A 350-gal spray tank covers 14 acres. Find the...Ch. 7.1 - Suppose 12 gal of herbicide concentrate are used...Ch. 7.1 - A flower bed contains a mixture of 6 ft3 of...Ch. 7.1 - Prob. 46ECh. 7.1 - Suppose 16 ft of copper tubing costs $27.04. Find...Ch. 7.1 - A structure has 3290 ft2 of wall area (excluding...Ch. 7.1 - A 2150-ft2 home sells for $268,750. Find the ratio...Ch. 7.1 - You need 15 ft3 of cement to make 80 ft3 of...Ch. 7.1 - A welder has 9 pieces of 4-ft steel angle and 12...Ch. 7.1 - A welder grabs a handful of 6011 welding rods and...Ch. 7.1 - The total number of hours required for a private-...Ch. 7.1 - Two small window air conditioner units were...Ch. 7.1 - A 1-litre (1000 mL) IV bag of dextrose solution...Ch. 7.1 - Over a period of 5 h, 1200 mL of an IV solution...Ch. 7.1 - Prob. 57ECh. 7.1 - Prob. 58ECh. 7.1 - Prob. 59ECh. 7.1 - In tilapia fish farming, 5 tons of high-quality...Ch. 7.1 - Find the flow rate for each given IV (assume a...Ch. 7.1 - Prob. 62ECh. 7.1 - Find the flow rate for each given IV (assume a...Ch. 7.1 - Find the flow rate for each given IV (assume a...Ch. 7.1 - Find the length of time each IV should be...Ch. 7.1 - Prob. 66ECh. 7.1 - Prob. 67ECh. 7.1 - Prob. 68ECh. 7.1 - Prob. 69ECh. 7.1 - Ann worked 6 hours and her pay was $54.60. What...Ch. 7.1 - Prob. 71ECh. 7.1 - Prob. 72ECh. 7.1 - A recipe calls for 1 gal of tomato paste and 3 qt...Ch. 7.1 - The soup of the day has water as 35% of its volume...Ch. 7.1 - Prob. 75ECh. 7.1 - 18 lb of potatoes are used in a recipe for 42...Ch. 7.2 - In each proportion, find a. the means, b. the...Ch. 7.2 - In each proportion, find a. the means, b. the...Ch. 7.2 - In each proportion, find a. the means, b. the...Ch. 7.2 - In each proportion, find a. the means, b. the...Ch. 7.2 - Prob. 5ECh. 7.2 - Prob. 6ECh. 7.2 - Determine whether or not each pair of rations is...Ch. 7.2 - Determine whether or not each pair of rations is...Ch. 7.2 - Determine whether or not each pair of rations is...Ch. 7.2 - Determine whether or not each pair of rations is...Ch. 7.2 - Determine whether or not each pair of rations is...Ch. 7.2 - Prob. 12ECh. 7.2 - Prob. 13ECh. 7.2 - Solve each proportion (round each result to three...Ch. 7.2 - Prob. 15ECh. 7.2 - Solve each proportion (round each result to three...Ch. 7.2 - Solve each proportion (round each result to three...Ch. 7.2 - Solve each proportion (round each result to three...Ch. 7.2 - Solve each proportion (round each result to three...Ch. 7.2 - Solve each proportion (round each result to three...Ch. 7.2 - Solve each proportion (round each result to three...Ch. 7.2 - Solve each proportion (round each result to three...Ch. 7.2 - Solve each proportion (round each result to three...Ch. 7.2 - Solve each proportion (round each result to three...Ch. 7.2 - Solve each proportion (round each result to three...Ch. 7.2 - Solve each proportion (round each result to three...Ch. 7.2 - Solve each proportion (round each result to three...Ch. 7.2 - Solve each proportion (round each result to three...Ch. 7.2 - Solve each proportion (round each result to three...Ch. 7.2 - Solve each proportion (round each result to three...Ch. 7.2 - Solve each proportion (round each result to three...Ch. 7.2 - Prob. 32ECh. 7.2 - Prob. 33ECh. 7.2 - Prob. 34ECh. 7.2 - Prob. 35ECh. 7.2 - Prob. 36ECh. 7.2 - Prob. 37ECh. 7.2 - Prob. 38ECh. 7.2 - Prob. 39ECh. 7.2 - Prob. 40ECh. 7.2 - Solve each proportion (round each result to three...Ch. 7.2 - Prob. 42ECh. 7.2 - You need 234ft3 of sand to make 8 ft3 of concrete....Ch. 7.2 - Prob. 44ECh. 7.2 - A builder sells an 1800-ft2 home for $171,000....Ch. 7.2 - Suppose 826 bricks are used in constructing a wall...Ch. 7.2 - Prob. 47ECh. 7.2 - A salesperson is paid a commission of $75 for...Ch. 7.2 - A plane flies for 3 h and uses 25 gal of 100LL...Ch. 7.2 - Metal duct that is 6 in. in diameter costs $7.50...Ch. 7.2 - Suppose 20 gal of water and 3 lb of pesticide are...Ch. 7.2 - A farmer uses 150 lb of a chemical on a 40-acre...Ch. 7.2 - Suppose a yield of 100 bu of corn per acre removes...Ch. 7.2 - Prob. 54ECh. 7.2 - An orchard has 3 rows with 5 apple trees each. The...Ch. 7.2 - A 25-lb bag of fertilizer covers 1200 ft2 of lawn....Ch. 7.2 - Prob. 57ECh. 7.2 - Prob. 58ECh. 7.2 - Prob. 59ECh. 7.2 - If welding rods cost $75 per 50 lb, how much would...Ch. 7.2 - An engine with displacement of 2.0 L develops 270...Ch. 7.2 - Prob. 62ECh. 7.2 - An automobile uses 18 gal of fuel to go 560 mi....Ch. 7.2 - A fuel pump delivers 35 mL of fuel in 420 strokes....Ch. 7.2 - A label reads: 2.5 mL of solution for injection...Ch. 7.2 - Prob. 66ECh. 7.2 - You are to administer 150 mg of aminophylline from...Ch. 7.2 - Prob. 68ECh. 7.2 - When finding the percent, round to the nearest...Ch. 7.2 - When finding the percent, round to the nearest...Ch. 7.2 - Prob. 71ECh. 7.2 - When finding the percent, round to the nearest...Ch. 7.2 - When finding the percent, round to the nearest...Ch. 7.2 - Prob. 74ECh. 7.2 - When finding the percent, round to the nearest...Ch. 7.2 - Prob. 76ECh. 7.2 - When finding the percent, round to the nearest...Ch. 7.2 - Prob. 78ECh. 7.2 - Prob. 79ECh. 7.2 - Prob. 80ECh. 7.2 - Prob. 81ECh. 7.2 - Prob. 82ECh. 7.2 - Find the number of millilitres of pure ingredient...Ch. 7.2 - Prob. 84ECh. 7.2 - Prob. 85ECh. 7.2 - Prob. 86ECh. 7.2 - Prob. 87ECh. 7.2 - Prob. 88ECh. 7.2 - Prob. 89ECh. 7.2 - Prob. 90ECh. 7.2 - The ratio of beef to pork for meat loaf is 5:2....Ch. 7.2 - Prob. 92ECh. 7.2 - The following recipe is for sherry vinegar...Ch. 7.2 - Prob. 94ECh. 7.2 - The following recipe is for 16 crepes: 14 cup...Ch. 7.2 - Prob. 96ECh. 7.2 - Prob. 97ECh. 7.2 - Prob. 98ECh. 7.2 - The kitchen ratio of ingredients for tenderloin of...Ch. 7.2 - Prob. 100ECh. 7.3 - Use the map in Figure 7.2 (page 266) to find the...Ch. 7.3 - Prob. 2ECh. 7.3 - Use the map in Figure 7.2 (page 266) to find the...Ch. 7.3 - Prob. 4ECh. 7.3 - Prob. 5ECh. 7.3 - Prob. 6ECh. 7.3 - Prob. 7ECh. 7.3 - Prob. 8ECh. 7.3 - Use the map in Figure 7.2 (page 266) to find the...Ch. 7.3 - Prob. 10ECh. 7.3 - Prob. 11ECh. 7.3 - Prob. 12ECh. 7.3 - Prob. 13ECh. 7.3 - Prob. 14ECh. 7.3 - Prob. 15ECh. 7.3 - Prob. 16ECh. 7.3 - Use the scale drawing of a metal plate cover in...Ch. 7.3 - Use the scale drawing of a metal plate cover in...Ch. 7.3 - Use the scale drawing of a metal plate cover in...Ch. 7.3 - Use the scale drawing of a metal plate cover in...Ch. 7.3 - Use the scale drawing of a metal plate cover in...Ch. 7.3 - Use the scale drawing of a metal plate cover in...Ch. 7.3 - Use the scale drawing of a metal plate cover in...Ch. 7.3 - Use the scale drawing of a metal plate cover in...Ch. 7.3 - Use the scale drawing of a metal plate cover in...Ch. 7.3 - Use the formulas for the hydraulic press to find...Ch. 7.3 - Use the formulas for the hydraulic press to find...Ch. 7.3 - Use the formulas for the hydraulic press to find...Ch. 7.3 - Prob. 36ECh. 7.3 - Use the formulas for the hydraulic press to find...Ch. 7.3 - Use the formulas for the hydraulic press to find...Ch. 7.3 - Prob. 39ECh. 7.3 - Use the formulas for the hydraulic press to find...Ch. 7.3 - Use the formulas for the hydraulic press to find...Ch. 7.3 - Prob. 42ECh. 7.3 - Prob. 43ECh. 7.3 - Use the formulas for the hydraulic press to find...Ch. 7.3 - Use the formulas for the hydraulic press to find...Ch. 7.3 - Use the formulas for the hydraulic press to find...Ch. 7.3 - Prob. 47ECh. 7.3 - Prob. 48ECh. 7.3 - Use the formulas for the hydraulic press to find...Ch. 7.3 - Use the formulas for the hydraulic press to find...Ch. 7.3 - Distance and time vary directly when driving at a...Ch. 7.3 - Distance and the amount of gasoline used vary...Ch. 7.3 - The number of patio blocks varies directly with...Ch. 7.3 - The number of people working on a project and the...Ch. 7.4 - Fill in the blanks: Pulley A Pulley B Diameter rpm...Ch. 7.4 - Fill in the blanks: Pulley A Pulley B Diameter rpm...Ch. 7.4 - Prob. 3ECh. 7.4 - Fill in the blanks: Pulley A Pulley B Diameter Rpm...Ch. 7.4 - Prob. 5ECh. 7.4 - Prob. 6ECh. 7.4 - Prob. 7ECh. 7.4 - Fill in the blanks: Pulley A Pulley B Diameter rpm...Ch. 7.4 - A small pulley is 13 in. in diameter, and a larger...Ch. 7.4 - A 21-in. pulley, rotating at 65 rpm, turns a...Ch. 7.4 - A large pulley turns at 48 rpm. A smaller pulley 8...Ch. 7.4 - Prob. 12ECh. 7.4 - A motor turning at 1870 rpm has a 4.0-in.-diameter...Ch. 7.4 - A hydraulic pump is driven with an electric motor...Ch. 7.4 - One pulley is 7 cm larger in diameter than a...Ch. 7.4 - One pulley is twice as large in diameter as a...Ch. 7.4 - Use Illustration 2 for Exercises 1718 ILLUSTRATION...Ch. 7.4 - Use Illustration 2 for Exercises 1718 ILLUSTRATION...Ch. 7.4 - Use Illustration 3 for Exercises 1920. Note:...Ch. 7.4 - Use Illustration 3 for Exercises 1920. Note:...Ch. 7.4 - Prob. 21ECh. 7.4 - Prob. 22ECh. 7.4 - Prob. 23ECh. 7.4 - Fill in the blanks: Gear A Gear B Number of teeth...Ch. 7.4 - Fill in the blanks: Gear A Gear B Number of teeth...Ch. 7.4 - Prob. 26ECh. 7.4 - A small gear with 25 teeth turns a large gear with...Ch. 7.4 - A large gear with 180 teeth running at 600 rpm...Ch. 7.4 - Prob. 29ECh. 7.4 - Prob. 30ECh. 7.4 - Prob. 31ECh. 7.4 - Prob. 32ECh. 7.4 - Complete the table F1 d1 F2 d2 18 lb 5 in. 9 lbCh. 7.4 - Prob. 34ECh. 7.4 - Prob. 35ECh. 7.4 - Prob. 36ECh. 7.4 - In Exercises 3337, draw a sketch for each and...Ch. 7.4 - A block of steel weighing 1800 lb is to be raised...Ch. 7.4 - A rocker arm raises oil from an oil well. On each...Ch. 7.4 - A carpenter needs to raise one side of a building...Ch. 7.4 - A 1200-g mass is placed 72 cm from the fulcrum of...Ch. 7.4 - Prob. 42ECh. 7.4 - Prob. 43ECh. 7.4 - A piece of machinery weighs 3 tons. It is to be...Ch. 7.4 - The length and width vary inversely in a rectangle...Ch. 7.4 - The speed and time of travel vary inversely when...Ch. 7.4 - Prob. 47ECh. 7.4 - Prob. 48ECh. 7 - Write each ratio in lowest terms: 7 to 28Ch. 7 - Write each ratio in lowest terms: 60:40Ch. 7 - Write each ratio in lowest terms: 1 g to 500 mgCh. 7 - Write each ratio in lowest terms: 5ft6in.9ftCh. 7 - Prob. 5RCh. 7 - Prob. 6RCh. 7 - Prob. 7RCh. 7 - Prob. 8RCh. 7 - Prob. 9RCh. 7 - Prob. 10RCh. 7 - Prob. 11RCh. 7 - Prob. 12RCh. 7 - Prob. 13RCh. 7 - Prob. 14RCh. 7 - A piece of cable 180 ft long costs $67.50. How...Ch. 7 - A copper wire 750 ft long has a resistance of 1.89...Ch. 7 - A crew of electricians can wire 6 houses in 144 h....Ch. 7 - An automobile braking system has a 12-to-1 lever...Ch. 7 - Jones invests $6380 and Hernandez invests $4620 in...Ch. 7 - Prob. 20RCh. 7 - Prob. 21RCh. 7 - Prob. 22RCh. 7 - Prob. 23RCh. 7 - Prob. 24RCh. 7 - Prob. 25RCh. 7 - Prob. 26RCh. 7 - Prob. 27RCh. 7 - An object 9 ft from the fulcrum of a lever...Ch. 7 - Prob. 29RCh. 7 - The number of workers needed to complete a...Ch. 7 - Prob. 1TCh. 7 - Prob. 2TCh. 7 - Write each ratio in lowest terms: 400 mL to 5 LCh. 7 - Prob. 4TCh. 7 - Prob. 5TCh. 7 - Prob. 6TCh. 7 - If 60 ft of fencing costs $138, how much does 80...Ch. 7 - Five quarts of pure antifreeze are added to 10...Ch. 7 - The scale on a map is 1 cm = 10 km. If two cities...Ch. 7 - A used automobile sells for $7500. The down...Ch. 7 - A small gear with 36 teeth turns a large gear with...Ch. 7 - A pulley is 20 cm in diameter and rotating at 150...Ch. 7 - Given the lever formula, F1d1=F2d2, and F1=800 lb,...Ch. 7 - A man who weighs 200 lb is to be raised by a lever...
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