To find the relationship between \( T \), \( P \), and \( V \), we can express the relationship mathematically. The problem states that \( T \) varies as the number of vehicles \( V \) in queue and inversely with the number of pumps \( P \). This can be expressed as:
\[ T = k \frac{V}{P} \]
where \( k \) is a constant of proportionality.
Step 1: Determine the Constant \( k \)
We are given that when there are 20 vehicles ( \( V = 20 \) ) and 5 pumps ( \( P = 5 \) ), the time taken is 10 minutes ( \( T = 10 \) ). We can substitute these values into the equation to find \( k \):
\[ 10 = k \frac{20}{5} \]
This simplifies to:
\[ 10 = k \cdot 4 \]
Thus,
\[ k = \frac{10}{4} = 2.5 \]
Now we can rewrite the relationship between \( T \), \( P \), and \( V \):
\[ T = 2.5 \frac{V}{P} \]
Step 2: Find the Time to Fuel 50 Vehicles with 2 Pumps
Next, we want to find the time \( T \) it will take to fuel 50 vehicles ( \( V = 50 \) ) with 2 pumps ( \( P = 2 \) ):
\[ T = 2.5 \frac{50}{2} \]
Calculating this gives:
\[ T = 2.5 \cdot 25 = 62.5 \text{ minutes} \]
Step 3: Find the Number of Pumps Required to Fuel 40 Vehicles in 20 Minutes
Now, we need to find the number of pumps \( P \) required to fuel 40 vehicles ( \( V = 40 \) ) in 20 minutes ( \( T = 20 \) ). We can rearrange the original equation to solve for \( P \):
\[ T = 2.5 \frac{V}{P} \implies P = 2.5 \frac{V}{T} \]
Substituting in \( V = 40 \) and \( T = 20 \):
\[ P = 2.5 \frac{40}{20} \]
This simplifies to:
\[ P = 2.5 \cdot 2 = 5 \]
Summary of Results:
- The relationship: \( T = 2.5 \frac{V}{P} \)
- Time to fuel 50 vehicles with 2 pumps: 62.5 minutes
- Number of pumps required to fuel 40 vehicles in 20 minutes: 5 pumps