To determine the proper relationship between the number of employees working on a task ([tex]$e$[/tex]) and the number of days it takes to complete the task ([tex]$d$[/tex]), we can analyze the information given and deduce a suitable mathematical model.
We are given two pieces of data:
1. It takes 4 employees 4 days to complete a task.
2. It takes 2 employees 8 days to complete the same task.
Let’s validate this relationship:
First data point:
[tex]\[
d_1 = 4 \text{ days}, \quad e_1 = 4 \text{ employees}
\][/tex]
Second data point:
[tex]\[
d_2 = 8 \text{ days}, \quad e_2 = 2 \text{ employees}
\][/tex]
We know that in both scenarios, each combination results in the completion of the same task.
Let’s see if the product of the number of days and the number of employees is constant in both scenarios:
[tex]\[
4 \text{ days} \times 4 \text{ employees} = 16
\][/tex]
[tex]\[
8 \text{ days} \times 2 \text{ employees} = 16
\][/tex]
Indeed, the product [tex]$d \cdot e$[/tex] remains the same and equals 16 in both cases. This suggests that the relationship between the number of employees and the number of days is an inverse variation, meaning:
[tex]\[
d \cdot e = k
\][/tex]
where [tex]$k$[/tex] is a constant.
Hence, the correct function to represent the relationship between [tex]$d$[/tex], the number of days, and [tex]$e$[/tex], the number of employees, is:
[tex]\[
\text{inverse variation}; \quad d \cdot e = k
\][/tex]
