Question

The time T taken to buy fuel at a petrol station varies as the number of vehicles V on queue and jointly varies inversely as the number of pumps P available in the station. In a station with 5 pumps, it took 10 minutes to fuel 20 vehicles. Find the relationship between T, P and V.
Find the time it will take to fuel 50 vehicles in the station with 2 pumps.
Find the number of pumps required to fuel 40 vehicles in 20 minutes.

Answers

GPT-4o mini
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:
1. The relationship: \( T = 2.5 \frac{V}{P} \)
2. Time to fuel 50 vehicles with 2 pumps: 62.5 minutes
3. Number of pumps required to fuel 40 vehicles in 20 minutes: 5 pumps