To solve this problem, we need to understand the relationship between the amount of time [tex]\( t \)[/tex] it takes to clean up the park and the number of volunteers [tex]\( v \)[/tex]. The problem states that [tex]\( t \)[/tex] varies inversely with [tex]\( v \)[/tex]. This means that when one increases, the other decreases proportionally.
The inverse variation relationship is expressed mathematically as:
[tex]\[ t \cdot v = k \][/tex]
where [tex]\( k \)[/tex] is a constant.
To determine this constant [tex]\( k \)[/tex], we use the given values: 7 volunteers ([tex]\( v = 7 \)[/tex]) and 1.5 hours ([tex]\( t = 1.5 \)[/tex]). Plugging these values into the equation gives us:
[tex]\[ k = t \cdot v = 1.5 \cdot 7 = 10.5 \][/tex]
Now, we have the constant [tex]\( k = 10.5 \)[/tex]. The equation that models this situation can now be written as:
[tex]\[ t = \frac{k}{v} \][/tex]
Substituting the value of [tex]\( k \)[/tex], we get:
[tex]\[ t = \frac{10.5}{v} \][/tex]
Thus, the equation that correctly models this situation is:
[tex]\[ \boxed{t = \frac{10.5}{v}} \][/tex]
Therefore, the correct answer is:
[tex]\[ \text{B. } t = \frac{10.5}{v} \][/tex]
