Question

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 100 workers 6 weeks to build 4 miles of highway. How many workers would be needed to build 16 miles of highway in 15 weeks?

Answer 1

Answer:

Therefore 160 workers needed to build 16 miles of highway in 15 weeks.

Step-by-step explanation:

Given that,

the amount of time to build a highway varies inversely with the number of workers and varies directly with the length of the highway.

i.e $T\propto\frac LW$

T= time, L= length of highway and W= number of workers.

So,we can rewrite the above equation

$\frac{T_1}{T_2}=\frac{L_1 W_2}{L_2W_1}$

Here, $W_1$= 100 worker, $L_1$ = 4 miles, $T_1$ = 15 weeks

$W_2$ = ?, $L_2$ = 16 miles, $T_2$ = 15 weeks

$\frac{T_1}{T_2}=\frac{L_1 W_2}{L_2W_1}$

$\Rightarrow \frac{6}{15}=\frac{4\times W_2}{16\times 100}$

$\Rightarrow \frac{6\times 16\times 100}{15\times 4}= W_2$         [ Multiply $\frac{16\times 100}{4}$ both sides]

$\Rightarrow W_2=\frac{6\times 16\times 100}{15\times 4}$

$\Rightarrow W_2=160$

Therefore to build 16 miles of highway in 15 weeks 160 workers needed.

Answer 2

The number of workers needed will be $160$.

Let

$t=\text{the amount of time taken to build the highway}\\l=\text{the length of the highway}\\n=\text{the number of workers}$

since the time taken is directly proportional to the length of the highway, but inversely proportional to the number of workers, we will have

$t\text{ }{\displaystyle \propto}\text{ } l \text{ and } t\text{ }{\displaystyle \propto}\text{ }n\\\implies t\text{ }{\displaystyle \propto}\text{ }\frac{l}{n}\\\implies t=\frac{kl}{n}$

substitute the values $t=6,l=4,n=100$ into the formula to find the proportionality constant $k$

$t=\frac{kl}{n}\\6=\frac{4k}{100}\\\\k=\frac{100\times 6}{4}=150$

now that we have our formula

$t=\frac{150l}{n}$

we can now solve our main problem by substituting the values $t=15,l=16$ to find $n$

$t=\frac{150l}{n}\\\\15=\frac{150\times16}{n}\\\\n=\frac{150\times16}{15}=160workers$

The number of workers needed to build a highway of $16\text{ miles}$ in $15\text{ weeks}$ will be $160$

Learn more about direct and inverse proportion here: https://brainly.com/question/1266676