Answer:
[tex]96.25[/tex] minutes.
Step-by-step explanation:
The question states that it takes the larger hose [tex]55[/tex] minutes to fill the entire pool on its own. Assume that the larger hose fills the pool at a constant rate. What fraction of the pool would this hose fill in each minute?
The larger hose fills [tex]\displaystyle \frac{1}{55}[/tex] of the pool in each minute. That is: [tex]\displaystyle \frac{1\; \text{pool}}{55\; \rm \text{minutes}} = \frac{1}{55}\; \text{pool} \cdot \text{minute}^{-1}[/tex].
Similarly, assume that the smaller hose is also filling the pool at a constant rate. The question states that it takes [tex]35[/tex] minutes for the two hoses to fill the pool when working together. Therefore, when working together, the two hoses would fill [tex]\displaystyle \frac{1}{35}[/tex] of the pool in each minute. That is: [tex]\displaystyle \frac{1\; \text{pool}}{35\; \rm \text{minutes}} = \frac{1}{35}\; \text{pool} \cdot \text{minute}^{-1}[/tex]
The difference between these two fractions should represent the fraction of the pool that the smaller hose fills in each minute:
The smaller hose fills [tex]\displaystyle \left( \frac{1}{35} - \frac{1}{55} \right)[/tex] of the pool in one minute. That's [tex]\displaystyle \left( \frac{1}{35} - \frac{1}{55} \right)\; \text{pool} \cdot \text{minute}^{-1}[/tex].
At [tex]\displaystyle \left( \frac{1}{35} - \frac{1}{55} \right)\; \text{pool} \cdot \text{minute}^{-1}[/tex], filling the entire pool would take
[tex]\displaystyle \frac{1\; \text{pool}}{\displaystyle \left( \frac{1}{35} - \frac{1}{55} \right)\; \text{pool} \cdot \text{minute}^{-1}} = \frac{1}{\displaystyle \frac{1}{35} - \frac{1}{55}}\; \text{minute} = 96.25\; \text{minute}[/tex].
Hence, it would take [tex]96.25[/tex] minutes for the smaller hose to fill the entire pool on its own.
