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].
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].
