MTH095 Intermediate Algebra Chapter 7 – Rational Expressions Sections 7.6 – Applications and Variations Motion (rate – time – distance) Shared Work Variation (direct, inverse, & joint) Copyright © 2010 by Ron Wallace, all rights reserved.
Motion Example … The current in the Lazy River moves at 4 mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water?
Motion Example … The current in the Lazy River moves at 4 mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water? Motion drt
Motion Example … The current in the Lazy River moves at 4 mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water? Motion drt Upstream
Motion Example … The current in the Lazy River moves at 4 mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water? Motion drt Upstream 6
Motion Example … The current in the Lazy River moves at 4 mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water? Motion drt Upstream 6r – 4
Motion Example … The current in the Lazy River moves at 4 mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water? Motion drt Upstream 6r – 4t
Motion Example … The current in the Lazy River moves at 4 mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water? Motion drt Upstream 6r – 4t Downstream
Motion Example … The current in the Lazy River moves at 4 mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water? Motion drt Upstream 6r – 4t Downstream 12
Motion Example … The current in the Lazy River moves at 4 mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water? Motion drt Upstream 6r – 4t Downstream 12r + 4
Motion Example … The current in the Lazy River moves at 4 mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water? Motion drt Upstream 6r – 4t Downstream 12r + 4t
Motion Example … The current in the Lazy River moves at 4 mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water? Motion drt Upstream 6r – 4t = 6/(r – 4) Downstream 12r + 4t = 12/(r + 4)
Motion Example … The current in the Lazy River moves at 4 mph. Monica’s dinghy motors 6 miles upstream in the same time it takes to motor 12 miles downstream. What would be the speed of her dinghy in still water? Time Up = Time Down
Motion In general … begin by filling in the table... Use a formula to eliminate a variable. Set equal expressions equal to each other. Solve & Check Motion drt Motion 1 ??? Motion 2 ???
Motion – Example A doctor drove 200 miles to attend a national convention. Because of poor weather, her average speed on the return trip was 10 mph less than her average speed going to the conventions. If the return trip took 1 hour longer, how fast did she drive in each direction? Motion drt Going there Coming home
Motion – Example A doctor drove 200 miles to attend a national convention. Because of poor weather, her average speed on the return trip was 10 mph less than her average speed going to the conventions. If the return trip took 1 hour longer, how fast did she drive in each direction? Motion drt Going there 200rt 1 = 200/r Coming home 200r – 10t 2 = 200/(r – 10)
Shared Work Example … Tom & Sue work for the city parks department where they mow the lawn in the city park. Tom, working by himself, can mow the lawn in 5 hours. Sue, working by herself, can mow the lawn in 4 hours. How long will it take to mow the lawn if they work together? Estimates? 9 hours? 4.5 hours? 2.25 hours? Other guesses?
Tom (working alone) takes 5 Hours
Hour #1
Tom (working alone) takes 5 Hours Hour #1 Hour #2
Tom (working alone) takes 5 Hours Hour #1 Hour #2 Hour #3
Tom (working alone) takes 5 Hours Hour #1 Hour #2 Hour #3 Hour #4
Tom (working alone) takes 5 Hours Hour #1 Hour #2 Hour #3 Hour #4 Hour #5
Sue (working alone) takes 4 Hours
Hour #1
Sue (working alone) takes 4 Hours Hour #1 Hour #2
Sue (working alone) takes 4 Hours Hour #1 Hour #2 Hour #3
Sue (working alone) takes 4 Hours Hour #1 Hour #2 Hour #3 Hour #4
Working Together Tom takes 5 hours. 1/5 of the job each hour. Sue takes 4 hours. 1/4 of the job each hour.
Working Together Hour #1 Hour #1 Tom takes 5 hours. 1/5 of the job each hour. Sue takes 4 hours. 1/4 of the job each hour.
Working Together Hour #1 Hour #1 Hour #2 Hour #2 Tom takes 5 hours. 1/5 of the job each hour. Sue takes 4 hours. 1/4 of the job each hour.
Working Together Tom takes 5 hours. 1/5 of the job each hour. Sue takes 4 hours. 1/4 of the job each hour. Hour #1 Hour #1 Hour #2 Hour #2 Hour #3 ?
Working Together Let x = # hours to complete the job together. Portion of work completed by Tom Portion of work completed by Sue Adding these gives … Tom takes 5 hours. 1/5 of the job each hour. Sue takes 4 hours. 1/4 of the job each hour.
Shared Work – Summary
Proportions – Equality of Ratios Ratio: A quotient of related quantities. Proportion: Two equivalent rations that relate the same quantities. One of the quantities will be unknown. Does it matter which quantity is on top?
Proportions – Equality of Ratios Example … An automobile gets 23 miles per gallon of gas (mpg). How much gas does it take to travel 200 miles? The Ratio …
Proportions – Equality of Ratios Example … An automobile gets 23 miles per gallon of gas (mpg). How much gas does it take to travel 200 miles? The Ratio …
Two Common Proportions Similar Triangles ◦ Corresponding Angles are Equal ◦ Ratios of Corresponding Sides are Equal Scale Drawings ◦ Maps ◦ Blueprints
Variation Direct Variation Two quantities whose ratio is a constant. Inverse Variation Two quantities whose product is a constant. Others – Combinations of the Above ◦ e.g. Joint Variation Three quantities where the ratio of one of the quantities to the product of the other two quantities is a constant.
Direct Variation Two quantities whose ratio is a constant. “ y varies directly as x ” aka: “ y is [directly] proportional to x ” k is called the “constant of proportionality” In an application, data is given to determine k and then values of x are used to determine values of y.
Direct Variation Example... The weight hanging from a spring is directly proportional to the distance the spring is stretched (Hooke’s Law). If a 6 pound weight stretches a particular spring 5 inches, and a fish hanging from the same spring stretches the spring 9 inches, how much does the fish weigh?
Inverse Variation Two quantities whose product is a constant. “ y varies inversely as x ” aka: “ y is inversely proportional to x ” k is called the “constant of proportionality” In an application, data is given to determine k and then values of x are used to determine values of y.
Inverse Variation Example... The time it takes a to get sunburned varies inversely with the UV rating on that day? If a UV rating of 4 causes person with fair skin to burn in 20 minutes, how long will it take for them to burn on a day with a UV rating of 7?