Written. APPLICATIONS OF EQUATIONS INDUSTRIAL PROBLEMS 1. In a factory there are three rates of pay: $1.25, $1.50, and $2 a day. There are 10 men more receiving $1.50 than $2.00, and 18 more receiving $1.25 than $1.50. The daily pay-roll amounts to $145; how many men are working at each rate? PLAN. 1. Let x be the number of men working for $2 a day. Why? men for $1.50, and men for $1.25. 2. A man inherited some money; he invested $500 at 4% and the rest at 5%; his income was $120 annually; how much did he inherit? 3. If an acre of land produces 20 tons of beets, and if beets yield 6% of their weight in sugar, how many acres of land are required to furnish the beets for a factory whose output of sugar is 190,000 lb. a year? 4. If a million pairs more of rubber shoes had been produced in the United States in 1900, there would have been one pair for every two persons of the 76 million inhabitants; find the number of pairs produced. 5. In a recent year the production of cane sugar in Cuba was twice that in the United States, 4 times that in Brazil, of that in the Sandwich Islands. As much was produced in Java as in the Sandwich Islands and the United States together. The total production in all of these countries was 2,400,000 tons. Find the production of each. Written. TRANSPORTATION PROBLEMS 195 1. A train with average speed 35 mi. per hour starts from New York for Philadelphia, 90 mi. distant, at the same time that a train with average speed 30 mi. an hour starts from Philadelphia for New York; when will the trains pass each other? PLAN. 1. Let x be the number of hr. after starting before they meet. 2. Then 35x is the distance traveled by the first train before 3. And 30x is what distance? 90. Why? 5. Therefore, x = —. they meet. 4. Therefore, 35x + 30x = 2. Two automobiles start from Chicago; one goes east at the rate of 20 mi. an hour and the other west at 15 mi. an hour; in how many hours are they 350 mi. apart? 3. A train 440 ft. long running at a speed of 20 mi. an hour requires 281 minutes to pass completely through the St. Gotthard tunnel in Switzerland; how long is the tunnel? 4. A train going from New York to Buffalo at the rate of 40 mi. an hour takes 2 hr. 12 min. longer than one going 50 mi. an hour; find the distance between these places? 5. The total number of miles of railway in the United States (to the nearest thousand) was 34,000 greater in 1900 than in 1890 and 5,000 greater than in 1899. The sum of all three numbers is 540,000. Find the mileage in 1900. 6. The number of freight locomotives in the United States in 1900 was 1,870 more than twice the number of passenger locomotives and 7,993 less than three times that number; find the number of each. 7. A baggage express company had 596 trunks to deliver. It delivered a certain number in the forenoon, of the whole in the afternoon and had 21 trunks undelivered night; how many were delivered before noon? 8. A ship set sail from San Francisco for Manila, 8,800 mi. distant. In 4 days the distance yet to sail was 7 times that already sailed. Find the time required to make the trip. 196 Written. PROBLEMS IN MEASUREMENT 1. The perimeter of a triangle is 38 in., the first side is 10 in. less than the second, and the third side is double the first. Find the length of each side. PLAN. 1. Let x be the length of the first side in inches. 2. Then x + 10 is the length of the second side in inches. 2. The total width of the American and Horseshoe Falls at Niagara is 3,700 ft. The Horseshoe Falls are 10 ft. less than 2 times as wide as the American Falls. Find the width of each. 3. The two highest structures in the world are the Eiffel Tower at Paris and the Washington Monument. The Eiffel Tower is 429 ft. higher than the monument, but lacks 126 ft. of being twice as high. Find the height of each. 4. The three highest stone structures in the world are the Washington Monument, the City Hall of Philadelphia, and the cathedral at Cologne, Germany. The first is 18 ft. higher than the second and 31 ft. higher than the third. The sum of their heights is 1,616 ft. Find the height of each. 5. A rug 12 ft. by 17 ft. covers of the floor of a room 18 ft. wide. Find the length of the room. PLAN. 1. Let L = the number of feet in length. 3. Therefore, L = 12 x 17. Why? 4. And L == 6. A rug 11 ft. by 15 ft. covers of the floor of a room 25 ft. long; find the breadth of the room. 7. A house 20 ft. by 52 ft. covers of a lot having 25 ft. frontage; find the depth of the lot. 8. Half the length of the Mississippi is 900 miles more than of its length; find its length. Written. PROBLEMS IN MEASUREMENT 197 1. Two brothers divided a farm of 160 acres so that one had 25 acres more than the other; how many acres had each? 2. The perimeter of a rectangle is 42 in. and the breadth is of the length; find the dimensions. 3. A picture requires 96 in. of molding to frame it. The length of the frame is of its breadth, outside measurement. Find the dimensions of the frame. 4. The breadth of a rectangular garden is of its length. The owner caused it to be covered 3 in. deep with black earth, costing him 60¢ a cubic yard. The expense was $15. Find the dimensions of the lot. 5. A rectangular park 100 yd. by 500 yd. is covered with lawns and flower beds; the lawn occupies 30,000 sq. yd. more than the flower beds; how many square yards are occupied by each? 6. One of the acute angles of a right-angled triangle is half the other; how many degrees are there in each? 7. One of the two equal sides of a triangle is 2 times the third side. The perimeter of the triangle is 21 ft. Find the lengths of the sides. 8. If a cubic foot of loose anthracite coal weighs 90 lb., how high must a bin 8 ft. by 10 ft. be to hold 30 tons? 9. Two boys measured the length of a playground. They took two sticks whose lengths they did not know, but saw that one stick was an inch longer than the other. One boy found the length of the playground to be 70 stick lengths, the other 68 stick lengths; what was the length of the playground? (Hint: Let x = length of shorter stick.) 10. The gauge of a water tank shows that it contains 620 gallons of water. The supply pipe furnishes 2 gallons per minute. At the end of 2 hours the gauge reads 560 gallons; what is the average consumption per minute? 198 PROBLEMS OF MOTION If a body moves so that it passes over equal spaces in equal intervals of time, the body is said to move uniformly. In the following problems uniform motion is meant. Written. 1. A man walked into the country at the rate of 3 mi. an hour. After spending 24 hr. there, he returned on the electric railway at the rate of 10 mi. an hour. His entire trip lasted 9 hours; how far did he go? 2. One wheel rider goes twice as fast as another. They start together and ride in the same direction about a circular track. In how many rounds will they pass for the first time? PLAN. 1. Let r be the number of rounds made by the slower rider until they pass. = 2. Then 2r the number of rounds made by the faster rider. 3. But the faster rider must go one round more than the slower, in order to overtake him. 4. Therefore, r+ 1 5. Test. 3. Solve the same problem if the riders go in opposite directions. 4. Solve the same problem if one rider goes 1 times as fast as the other in the same direction. 5. Solve Exercise 4, the riders going in opposite directions. 6. A train with an average speed of 30 mi. per hour leaves Chicago at noon. At 2 P.M. a second train, with an average speed of 40 mi. per hour, leaves on the same line and in the same direction. When will the second train overtake the first? 7. A man set out to ride a distance of 54 miles. His horse trotted 6 mi. per hour. After he had gone a certain distance, his horse was lamed, and he had to proceed afoot, walking 3 mi. per hour. On arriving at his destination he found that he had ridden and walked an equal length of time. How many hours did it take him to make the journey? |