Two hoses attached to separate water sources are available to fill a cyindrical swimming pool. If both hoses are used , the time it will take to fill the pool can be represented by the following equation : `(1)/(T _(1))+(1)/(T_(2))=(1)/(T_(c))`, where `T_(1) and T_(2)` represent the time needed for hoses 1 and 2 , respectively , to fill the pool on their own , and `T_(c) ` represents the time needed for hoses 1 and 2 to fill the pool working together . IF hose 1 alone can fill the pool in exactly 20 minutes and hose 2 alone can fill the pool in exactly 60 minutes , how many minutes will it take to fill the pool if both hoses work simultaneously ?

Options

A. 3
B. 10
C. 15
D. 18

Answer

To solve the problem, we will use the equation provided in the question:

\[ \frac{1}{T_1} + \frac{1}{T_2} = \frac{1}{T_c} \]

where:
- \(T_1\) is the time taken by hose 1 to fill the pool on its own,
- \(T_2\) is the time taken by hose 2 to fill the pool on its own,
- \(T_c\) is the time taken by both hoses to fill the pool together.

### Step 1: Identify the values of \(T_1\) and \(T_2\)
From the problem, we know:
- Hose 1 can fill the pool in \(T_1 = 20\) minutes.
- Hose 2 can fill the pool in \(T_2 = 60\) minutes.

### Step 2: Substitute the values into the equation
Now, we substitute \(T_1\) and \(T_2\) into the equation:

\[ \frac{1}{20} + \frac{1}{60} = \frac{1}{T_c} \]

### Step 3: Calculate the left-hand side
To add the fractions \(\frac{1}{20}\) and \(\frac{1}{60}\), we need a common denominator. The least common multiple (LCM) of 20 and 60 is 60.

Rewriting \(\frac{1}{20}\) with a denominator of 60:

\[ \frac{1}{20} = \frac{3}{60} \]

Now we can add:

\[ \frac{3}{60} + \frac{1}{60} = \frac{4}{60} \]

### Step 4: Simplify the fraction
Now simplify \(\frac{4}{60}\):

\[ \frac{4}{60} = \frac{1}{15} \]

### Step 5: Set the equation for \(T_c\)
Now we have:

\[ \frac{1}{T_c} = \frac{1}{15} \]

### Step 6: Solve for \(T_c\)
Taking the reciprocal of both sides gives us:

\[ T_c = 15 \]

### Conclusion
Thus, it will take **15 minutes** to fill the pool if both hoses work simultaneously.

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