Mathematics Assignment

(1) The time [tex]\((t)\)[/tex] taken to buy fuel at a filling station varies directly as the number of vehicles [tex]\((v)\)[/tex] in a queue and varies inversely as the number of pumps [tex]\((p)\)[/tex] available at the station. In a station with 5 pumps, it took 20 minutes to fuel 20 vehicles.

Find the:
(a) Relationship between [tex]\(t\)[/tex], [tex]\(v\)[/tex], and [tex]\(p\)[/tex].



Answer :

To solve this problem, we start with the given mathematical relationship:

1. The time [tex]\( t \)[/tex] taken to buy fuel at a filling station varies directly as the number of vehicles [tex]\( \omega \)[/tex] in a queue. This implies:
[tex]\[ t \propto \omega \][/tex]

2. The time [tex]\( t \)[/tex] varies inversely as the number of pumps [tex]\( p \)[/tex] available at the station. This implies:
[tex]\[ t \propto \frac{1}{p} \][/tex]

Combining these two relationships, we get:
[tex]\[ t \propto \frac{\omega}{p} \][/tex]

We can express this as:
[tex]\[ t = k \cdot \frac{\omega}{p} \][/tex]
where [tex]\( k \)[/tex] is the constant of proportionality.

Next, we need to find the value of [tex]\( k \)[/tex] using the given information:

- Number of pumps, [tex]\( p = 5 \)[/tex]
- Number of vehicles, [tex]\( \omega = 20 \)[/tex]
- Time taken, [tex]\( t = 10 \)[/tex] minutes (noting that "comm" appears to be a typo for "10 minutes")

Using these values, we substitute into the equation:
[tex]\[ 10 = k \cdot \frac{20}{5} \][/tex]

Simplifying the right-hand side:
[tex]\[ 10 = k \cdot 4 \][/tex]

Solving for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{10}{4} \][/tex]
[tex]\[ k = 2.5 \][/tex]

Therefore, the relationship between the time [tex]\( t \)[/tex], the number of vehicles [tex]\( \omega \)[/tex], and the number of pumps [tex]\( p \)[/tex] is:
[tex]\[ t = 2.5 \cdot \frac{\omega}{p} \][/tex]

So, the proportional relationship is:
[tex]\[ t = 2.5 \cdot \frac{\omega}{p} \][/tex]

This equation succinctly describes the time required to fuel vehicles at the station, considering it varies directly with the number of vehicles and inversely with the number of pumps available.