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Suppose that the amount of time it takes to build a highway varies directly with the length of the highway and inversely with the number of workers. Suppose also that it takes 300 workers 22 weeks to build 24 miles of highway. How long will it take 225 workers to build 27 miles of highway?
◻weeks

Suppose that the amount of time it takes to build a highway varies directly with the length of the highway and inversely with the number of workers. Suppose also that it takes $300$ workers $22$ weeks to build $24$ miles of highway. How long will it take $225$ workers to build $27$ miles of highway? $\newline$ $\square$ weeks

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Evaluate two-variable equations: word problems

Full solution

Q. Suppose that the amount of time it takes to build a highway varies directly with the length of the highway and inversely with the number of workers. Suppose also that it takes

300

workers

22

weeks to build

24

miles of highway. How long will it take

225

workers to build

27

miles of highway?

\newline

\square

weeks

Define Variables: Let's denote the time it takes to build the highway as $T$ , the number of workers as $W$ , and the length of the highway as $L$ . According to the problem, $T$ varies directly with $L$ and inversely with $W$ . This relationship can be expressed as $T = k \times (L / W)$ , where $k$ is the constant of variation.
Calculate Constant of Variation: We are given that it takes $300$ workers $22$ weeks to build $24$ miles of highway. We can use this information to find the constant of variation $k$ . Plugging in the values, we get $22 = k \times (24 / 300)$ .
Find Time for $225$ Workers: To find $k$ , we solve the equation: $k = \frac{22}{\frac{24}{300}} = 22 \times \left(\frac{300}{24}\right) = 22 \times 12.5 = 275$ .
Find Time for $225$ Workers: To find $k$ , we solve the equation: $k = \frac{22}{\frac{24}{300}} = 22 \times \frac{300}{24} = 22 \times 12.5 = 275$ .Now that we have the constant of variation $k$ , we can use it to find out how long it will take $225$ workers to build $27$ miles of highway. We plug in the values into the formula $T = k \times \frac{L}{W}$ , getting $T = 275 \times \frac{27}{225}$ .
Find Time for $225$ Workers: To find $k$ , we solve the equation: $k = \frac{22}{\frac{24}{300}} = 22 \times \frac{300}{24} = 22 \times 12.5 = 275$ .Now that we have the constant of variation $k$ , we can use it to find out how long it will take $225$ workers to build $27$ miles of highway. We plug in the values into the formula $T = k \times \frac{L}{W}$ , getting $T = 275 \times \frac{27}{225}$ .We calculate $T$ : $T = 275 \times \frac{27}{225} = 275 \times 0.12 = 33$ weeks.

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