The amount of time [tex][tex]$t$[/tex][/tex] it takes for a group of volunteers to clean up a park varies inversely with the number of volunteers [tex][tex]$v$[/tex][/tex]. If it takes 7 volunteers 1.25 hours to clean up the park, which of the following equations models this situation?

A. [tex]t = \frac{8.25}{v}[/tex]
B. [tex]t = \frac{8.75}{v}[/tex]
C. [tex]t = \frac{7.25}{v}[/tex]
D. [tex]t = \frac{5.75}{v}[/tex]



Answer :

To determine which equation correctly models the situation where the time [tex]\( t \)[/tex] it takes to clean up a park varies inversely with the number of volunteers [tex]\( v \)[/tex], we need to follow these steps:

1. Understand inverse variation: When two variables vary inversely, their product is constant. In this case, the time [tex]\( t \)[/tex] and the number of volunteers [tex]\( v \)[/tex] vary inversely, so [tex]\( t \cdot v \)[/tex] is a constant.

Mathematically, [tex]\( t \cdot v = k \)[/tex], where [tex]\( k \)[/tex] is the constant of variation.

2. Use the given values: We are given that it takes 7 volunteers [tex]\( v = 7 \)[/tex] to clean up the park in 1.25 hours [tex]\( t = 1.25 \)[/tex].

3. Calculate the constant [tex]\( k \)[/tex]:
[tex]\[ k = t \cdot v = 1.25 \cdot 7 = 8.75 \][/tex]

4. Formulate the equation: Since [tex]\( t \cdot v = 8.75 \)[/tex], we can write [tex]\( t \)[/tex] in terms of [tex]\( v \)[/tex]:
[tex]\[ t = \frac{8.75}{v} \][/tex]

5. Compare with the given options:
- A. [tex]\( t = \frac{8.25}{v} \)[/tex]
- B. [tex]\( t = \frac{8.75}{v} \)[/tex]
- C. [tex]\( t = \frac{7.25}{v} \)[/tex]
- D. [tex]\( t = \frac{5.75}{v} \)[/tex]

From our calculation, the correct equation that models the situation is:
[tex]\[ t = \frac{8.75}{v} \][/tex]

Thus, the correct option is [tex]\( \boxed{B} \)[/tex].

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