<p>An application of a rational function is T = (AB)/(A+B), which gives the time
ID: 3106359 • Letter: #
Question
<p>An application of a rational function is T = (AB)/(A+B), which gives the time, T, it takes for two workers to complete a particular task where A & B represent the time it would take for each individual worker to complete the identical task working alone. It takes William 3 hours longer than Timothy to paint a 4x 5 feet bedroom. Respond to the following:</p><ol>
<li>Working together, both can complete the job in 2 hours. How long does it take each one to complete the painting job working alone?<strong> </strong></li>
<li>Provide a similar example, set up the equation and solve the problem.</li>
<li>Show you work step by step and explain your work step by step.</li>
<li>You should discuss the problem as if you tutor your classmates.</li>
</ol>
Explanation / Answer
The first step here is to realize that William and Timothy can be represented by the variables A and B in the equation T = (AB)/(A+B).
So, lets say that hours worked by William is A and hours worked by Timothy is B. The problem tells you that William takes 3 hours more than Timothy or, in other words, A takes 3 hours more than B. You can state this mathmatically by writing the following:
A = B + 3
Since it is easier to solve an equation with one variable than one with two, and we know that A = B + 3, we can put B + 3 into the first equation wherever we see an A. This gives us:
([B+3]B)/(B+B)
Simpifying the equation gives us
([B+3]B)/(B+B) = ([B+3]B)/(2B) = (B+3)/2
The problem then states that the problem takes them 2 hours working together, so we set the equation we just got equal to 2 hours and solve for B.
(B+3)/2 = 2... (B+3) = 4... B = 1.
Earlier we said that B represented the hours woked by Timothy, and just above we found that B = 1. So the amount of hours worked by Timothy alone is 1 hour.
To find the hours worked by William we just go back to the equation that says that William takes 3 hours more than Timothy and substitute 1 in everywhere there is a B. We find that:
A = B + 3 = 1 + 3 = 4
So A = 4, or the amount of hours worked by William alone is 4 hours.
A simial example would be something like the time it takes two workers to finish building a ramp is described by the equation, T = (AB+1)/2. The first worker, A in this equation, takes 3 hours to build the ramp alone and the ramp can be built in 2 hours if the two work together. How long does it take the second worker, B in this equation, to build the ramp alone.
In this problem we are given that the value of T is 2. We are also given that A is 3. Since we are given both numbers we just substitute 2 in for every T and 3 in for every A.
T = (AB+1)/2... 2 = (3B+1)/2
When we simplifiy the equation we get:
(3B+1)/2 = 2... (3B+1) = 4... 3B = 3... B = 1.
Since we are given that B = 1, we know that the second worker would only take 1 hour to complete the ramp.
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