Question 2
The time (t) taken to buy a fuel at a petrol station variation varies directly as the number of vechice on a quene and inwersely as the number of pumps § available in a station. In a station of 5 pumps, it took 10 minutes to fuel 20 vechiles
Find;
(a) The relationship betweet and .
(b) The time it will taple to fuel 50 vechiles in Station usithe 2 pumps

Ask by Bright Montgomery. in Ghana

Jan 23,2025

Question

Question 2
The time (t) taken to buy a fuel at a petrol station variation varies directly as the number of vechice on a quene and inwersely as the number of pumps § available in a station. In a station of 5 pumps, it took 10 minutes to fuel 20 vechiles
Find;
(a) The relationship betweet and .
(b) The time it will taple to fuel 50 vechiles in Station usithe 2 pumps

Ask by Bright Montgomery. in Ghana

Jan 23,2025

UpStudy AI · Accepted Answer

Let's solve the problem step by step.

### Given:
- **Direct Variation:** Time $$ t $$ varies directly with the number of vehicles $$ v $$.
- **Inverse Variation:** Time $$ t $$ varies inversely with the number of pumps $$ p $$.

### (a) Establishing the Relationship Between $$ t $$, $$ p $$, and $$ v $$

When a variable varies **directly** with one variable and **inversely** with another, the relationship can be expressed as:

$$
t = k \times \frac{v}{p}
$$

Where:
- $$ t $$ = Time taken to fuel the vehicles
- $$ v $$ = Number of vehicles in the queue
- $$ p $$ = Number of pumps available
- $$ k $$ = Constant of proportionality

**So, the relationship is:**

$$
t = \frac{k \cdot v}{p}
$$

### (b) Calculating the Time to Fuel 50 Vehicles with 2 Pumps

**Step 1: Find the Constant $$ k $$ Using Given Data**

Given:
- $$ p = 5 $$ pumps
- $$ v = 20 $$ vehicles
- $$ t = 10 $$ minutes

Plug these values into the relationship:

$$
10 = \frac{k \cdot 20}{5}
$$

$$
10 = 4k
$$

$$
k = \frac{10}{4} = 2.5
$$

**Step 2: Use $$ k $$ to Find the Time for $$ v = 50 $$ Vehicles and $$ p = 2 $$ Pumps**

Plug $$ k = 2.5 $$, $$ v = 50 $$, and $$ p = 2 $$ into the relationship:

$$
t = \frac{2.5 \cdot 50}{2}
$$

$$
t = \frac{125}{2} = 62.5 \text{ minutes}
$$

**Therefore, it will take **62.5 minutes** to fuel 50 vehicles using 2 pumps.**

### Summary

- **(a)** The relationship is $$ t = \frac{k \cdot v}{p} $$.
- **(b)** It will take **62.5 minutes** to fuel 50 vehicles with 2 pumps.
(a) The relationship is $$ t = \frac{k \cdot v}{p} $$. (b) It will take 62.5 minutes to fuel 50 vehicles with 2 pumps.

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