21. A work crew can build a wall in 6 hours. If a second crew joins, the time to build the wall is reduced by half. What rational function represents the total time T(x), where x is the number of work crews?
Question 21Answer

a.
T(x)=6/x

b.
T(x)=x6

c.
T(x)=6x2 ​

d.
T(x)=12x

Question:A work crew can build a wall in 6 hours. If a second crew joins, the time to build the wall is reduced by half. What rational function represents the total time T(x), where x is the number of work crews?Solution Process:Let the amount of work to build the wall be W.One work crew can build the wall in 6 hours, so the rate of one crew is [tex]R_{1} =\frac{W}{6}[/tex].When a second crew joins, the time to build the wall is reduced by half, which means it takes 3 hours. So, with two crews, the combined rate is [tex]R_{2} =\frac{W}{3}[/tex].Let's assume each crew works at the same rate. If x is the number of work crews, the total rate is  [tex]x * \frac{W}{6}[/tex].The time to build the wall is [tex]T(x) = \frac{W}{x*\frac{W}{6} } =\frac{6}{x}[/tex].However, this doesn't account for the given information that two crews take half the time. Let r be the rate of one crew. So, [tex]r=\frac{1}{6}[/tex] walls per hour.If a second crew joins, the time is reduced by half, meaning it takes 3 hours. The combined rate is [tex]\frac{1}{3}[/tex] walls per hour.Let's say the rate of the second crew is [tex]r_{2}[/tex]. Then [tex]r+r_{2} =\frac{1}{3}[/tex].[tex]\frac{1}{6}+r_{2}=\frac{1}{3}[/tex][tex]r_{2}=\frac{1}{3}-\frac{1}{6}=\frac{2}{6}-\frac{1}{6}=\frac{1}{6}[/tex].So, the crews work at the same rate.If x is the number of work crews, then the combined rate is [tex]x * \frac{1}{6}[/tex].The time to build the wall is [tex]T(x)=\frac{1}{x*\frac{1}{6} }=\frac{6}{x}[/tex].Answer:a. [tex]T(x)=\frac{6}{x}[/tex]

Answered by Velvrix | 2025-08-21