Question

Respond to the following prompts in a minimum of 175 words: How do you think rational expressions will be applied in your future career? Provide an example in your explanation.

Answer 1

Rational expressions and rational equations can be useful tools for representing real life situations and for finding answers to real problems. In particular, they are quite good for describing distance-speed-time questions, and modeling multi-person work problems. Solving Work Problems Work problems often ask us to calculate how long it will take different people working at different speeds to finish a task. The algebraic models of such situations often involve rational equations derived from the work formula, W = rt. The amount of work done (W) is the product of the rate of work (s) and the time spent working (t). The work formula has 3 versions: W=1t W 1- w .. Some work problems have multiple machines or people working on a project together for the same amount of time but at different rates. In that case, we can add their individual work rates together to get a total work rate. Let's look at an example: Problem Example Myra takes 2 hours to plant 500 power bulbs. Francis takes 3 hours to plant 450 nower bulbs. Working together, how long should It take them to plant 1500 bulbs? Myra 500 bulbs/2 hours = 250 Think about how many bulbs each bulbs/hour person can plant in one hour. This is their planting rate Francis: 450 bulbs/3 hours = 150 bulbs/hour Myra & Francis together Combine their hourly rates to 250 + 150 bulbs/hour = 400 determine the rate they work bulbs/hour together 400 1500 1 Use one of the work formulas to write a rational equation, for W example=We know r, the combined work rate, and we know W, the amount of work that must be done What we don't know is how much time it will take to do the required work at the designated rate 400 1500 Solve the equation by multiplying both sides by the common denominator, then isolating t 4001 - 1500 = 1500 15 400 4 = t=3 hours Answer It takes 3 hours 45 minutes for Myra and Francis to plant 1500 bulbs together Other work problems go the other way. We'll calculate how long it will take one person to do a job alone when we know how long it takes a group to get it done: Problem Pria + Paul + Jamie = Example Jamie, Pria and Paul can paint a room together in 2 hours. If Pria does the Job alone she can paint the room in 5 hours. If Paul works alone, he can paint the room in 6 hours. If Jamie works alone, how long would it take her to paint the room? Determine the hourly rates for each room/hour person and for the whole group using w the formula lī Pria room/hour Work is painting 1 room, so W = 1 ܝ ܙܗ Paul 01- room/hour We don't know how long Jamie will take, so we need to keep the variablet. Jamie -- room/hour 1 1 Write an equation to show that the sum of their individual rates equals the group rate (Think of it this way, Pria works for one 1 hour and paints of the room. Paul works for an hour and paints of the room Jamie works for an hour and 1 paints ; of the room. Together they have painted half the room in one hour. Solve the rational equation First find the least common denominator of the individual rates. It is 5.6.1 = 30t. 1.61 151 1.30 1 5 616 51 302 Then multiply each term on the left by a fractional form of 1 so that all rates have the same denominator and can be added (Note: we could also have found the common denominator of the entire equation, which is also 30t and multiplied both sides by it) 61 51 30 1 301 301 3012 + 111+30 1 301 2 111 +30 301 2.30 = 2:30 301 Now multiply both sides of the equation by the common denominator, then simplify 111 + 30 111 + 30 =15 111 - 111+30 - 15t-11t 30 = 4 7.5 = It would take Jamie 7.5 hours to paint the room by herself. Answer Sometimes work problems describe rates in a relative way: someone works 3 times as fast as someone else or a machine takes 2 fewer hours to finish a job than another model of machine. In these instances, we express one rate using information about another rate. Let's look at an example: Problem Example One pipe can nil a pool 1.5 times faster than a second pipe. If both pipes are open, the pool can be filled in 6 hours. If only the slower pipe is open, how long would it take to Nll the pool? w Find the rates of each pipe alone 1 and the two working together Work = 1 pool filled 1 fast pipe - Hours needed for fast pipe to fil p pool p Hours needed for slow pipe to fill 1 pool alone 1.5 p slow pipe - 1.5p Hours needed for both pipes together: 6 fast pipe + slow pipe = 1 6 1 1 1 P 15p 6 Write an equation that shows that the amount of work completed by both pipes in one hour is equal to the sum of the work of each pipe. Solve for p. One way to do this is to rewrite the rational expressions using a common denominator, 1.6 4 1.2 1 p 6 1.5p 4 6 p Common denominator of p, 1.5p and 6 is Ep. 6 4 р 6p 6p 6p 6+4 =p p=10 hours 10.1.5 - 15 hours Slow pipe takes 1.5p hours to do the work alone Answer The slower pipe will take 15 hours to fill the pool alone Summary Rational equations can be used to solve a variety of problems that involve rates, times and work. Using rational expressions and equations can help us answer questions about how to combine workers or machines to complete a job on schedule. Scanned with CamScanner CS Scanned with CamScanner