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# applications of rational equations word problems

## applications of rational equations word problems

This divisional form leads to rational equations. Sam can paint a house in 5 hours. 36. $\displaystyle \frac{V}{\pi {{r}^{2}}}=h$, $\displaystyle h=\frac{V}{\pi {{r}^{2}}}$. Alex builds the same small shed in 2 days. Working separately, how long does it take each crew to build a shed? Manny paints the office in 10 hours. $\large \begin{array}{l}1=\left( \frac{1}{x}+\frac{1}{3x} \right)24\\\\1=\left[ \frac{\text{1}}{\text{32}}+\frac{1}{3\text{(32})} \right]24\\\\1=\frac{24}{\text{32}}+\frac{24}{3\text{(32})}\\\\1=\frac{24}{\text{32}}+\frac{24}{96}\\\\1=\frac{3}{3}\cdot \frac{24}{\text{32}}+\frac{24}{96}\\\\1=\frac{72}{96}+\frac{24}{96}\end{array}$. Similarly, when the unknown quantity is the rate, the setup also may result in a rational equation. 34. On the return trip, he was able to average 14 miles per hour faster than he averaged on the trip to town. Because the trains travel the same amount of time, finish the algebraic setup by equating the expressions that represent the times: Solve this equation by first multiplying both sides by the LCD, x(x+20). When solving distance problems where the time element is unknown, use the equivalent form of the uniform motion formula. Using rational expressions and equations can help you answer questions about how to combine workers or machines to complete a job on schedule. Use this information to set up an algebraic equation that models the application. To avoid introducing two more variables for the time column, use the formula t=Dr. These unique features make Virtual Nerd a viable alternative to private tutoring. Example 5: Mary spent the first 120 miles of her road trip in traffic. Use algebra to solve the following applications. The algebraic models of such situations often involve rational equations derived from the work formula, $W=rt$. Francis takes 3 hours to plant 45 flower bulbs. John thinks that if he worked alone, it would take him 3 times as long as it would take Joe to paint the entire house. Rowing downstream, the current increases his speed, and his rate is x + 2 miles per hour. Working together, they clean the carpets in 6 hours. Jeremy can paint the same shed by himself in 8 hours. Use the formula t=Dr to fill in the time column for each train. Find the integers. The algebraic models of such situations often involve rational equations derived from the work formula, $W=rt$. How long would it take Bryan to prepare and paint the house by himself? 30. Solve the equation by multiplying both sides by the common denominator, then isolating $t$. 3. Rowing upstream, the current decreases his speed, and his rate is x − 2 miles per hour. If the sum of the reciprocal of the smaller and twice the reciprocal of the larger is 17/35, then find the two integers. On the return trip against the wind, it covers 190 miles in the same amount of time. If the train was 18 miles per hour faster than the bus and the total trip took 2 hours, then what was the average speed of the train? How long will it take them to paint two sheds working together? Answer: His rowing speed is 6 miles per hour. 42. Think about how many bulbs each person can plant in one hour. Sally was able to drive an average of 20 miles per hour faster in her car after the traffic cleared. $\displaystyle \frac{40}{1}=\frac{150}{t}$. Multiply the individual work rates by 2 hours to fill in the chart. How long would it take her associate to complete the route by herself? In our last example we will define an equation that models the concentration  – or ratio of sugar to water – in a large mixing tank over time. A large mixing tank currently contains 100 gallons of water into which 5 pounds of sugar have been mixed. In that case, you can add their individual work rates together to get a total work rate. Ex 2: Solve a Literal Equation for a Variable. How long will it take them to detail a car working together? The rate at which a task can be performed. The algebraic models of such situ… As you will see, if you can find a formula, you can usually make sense of a situation. Many other application problems require finding an average value in a similar way, giving us variables in the denominator. In this non-linear system, users are free to take whatever path through the material best serves their needs. Working together, they take inventory in 1.5 hours. What was his average speed on the trip to town? 47. If the total trip took 9 hours, then how fast was she moving in traffic? Review on Rational Equations and Word Problems Solve the following equations. Write an expression to represent each person’s rate using the formula $\displaystyle r=\frac{W}{t}$. Combine their hourly rates to determine the rate they work together. Some work problems include multiple machines or people working on a project together for the same amount of time but at different rates. If Jerry starts the job and his assistant joins him 1 hour later, then how long will it take to lay the floor? If the total trip took 2 hours, then what was her average speed in traffic? If the sum of the reciprocal of the smaller and twice the reciprocal of the larger is 5/12, then find the two integers. Gary can do it in 4 hours. Working alone, Monique takes 4 hours longer than Audrey to record the inventory of the entire shop. Solution: The given information tells us that Billy’s dad has an individual work rate of 13 task per hour. 33. Solving Application Problems . 14. 11. 19: Trolley: 30 miles per hour; bus: 36 miles per hour, 21: Passenger car: 66 miles per hour; aircraft: 130 miles per hour. \end{array}\\ & & \\ \begin{array}{|c|c|c|c|} \hline & \,\text{rate}\,& \,\text{time}\,& \,\text{distance}\\ \hline \,\text{There}\,& r & t & 120\\ \hline \,\text{Back}\,& r - 10 & t + 2 & 120\\ \hline \end{array}& & \begin{array}{l} \,\text{Coming}\,\,\text{back}\,\,\text{he}\,\,\text{drove}\,10 \,\text{mph} \,\text{slower}\,(r - 10)\\ \,\text{and}\,\,\text{took}\,2 \,\text{hours}\,\,\text{longer}\,(t + 2) . How long would it take his assistant to assemble a computer working alone? By the end of this chapter, students should be able to:  Identify extraneous values  Apply methods of solving rational equations to solve rational equations  Solve applications with rational equations including revenue, distance, and work-rate problems Using rational expressions and equations can help you answer questions about how to combine workers or machines to complete a job on schedule. The fixed cost doesn’t change when more items are produced, whereas the variable cost increases as more cars are produced. A “work problem” is an example of a real life situation that can be modeled and solved using a rational equation. The sum of the portions of the task results in the total amount of work completed. Here is the guiding principle. When solving problems using rational formulas, it is often helpful to first solve the formula for the specified variable. A positive integer is twice another. $\displaystyle \frac{V}{\pi {{r}^{2}}}=\frac{\pi {{r}^{2}}h}{\pi {{r}^{2}}}$. Of course, the unit of time for the work rate need not always be in hours. If Billy helps his dad, then the yard work takes 2 hours. 15. What is the speed of the aircraft in calm air? Using rational expressions and equations can help you answer questions about how to combine workers or machines to complete a job on schedule.