Question 2 The time ( $$ t $$ ) taken to buy a fuel at a petrol station Variation varies directly as The number of vechice $$ v $$ on a quenc and inversely as the number of pumps (p) available in a station. In a station of 5 pumps, it took 10 minutes to fuel 20 vechiles Find; (a) The relationship betweet $$ t $$ $$ p $$ and $$ U $$. (b) The time it will tablie to puel 50 vechiles in station usitte 2 pumps

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Let's address each part of the question step by step.

### **Given:**
- **Time taken to buy fuel ($$ t $$)** varies **directly** as the **number of vehicles ($$ v $$)** on a queue.
- $$ t $$ varies **inversely** as the **number of pumps ($$ p $$)** available at the station.
- In a station with **5 pumps**, it took **10 minutes** to fuel **20 vehicles**.

---

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

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

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

Where:
- $$ k $$ is the **constant of proportionality**.

**Steps to Determine the Relationship:**
1. **Express the proportionality:**
   $$
   t \propto \frac{v}{p}
   $$
2. **Introduce the constant $$ k $$:**
   $$
   t = k \times \frac{v}{p}
   $$

**Final Relationship:**
$$
\boxed{t = k \cdot \frac{v}{p}}
$$

---

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

**Given:**
- Number of pumps ($$ p $$) = 2
- Number of vehicles ($$ v $$) = 50

**First, determine the constant $$ k $$ using the initial conditions:**
- $$ t = 10 $$ minutes when $$ v = 20 $$ and $$ p = 5 $$.

$$
10 = k \times \frac{20}{5} \
10 = k \times 4 \
k = \frac{10}{4} \
k = 2.5
$$

**Now, use the value of $$ k $$ to find the new time ($$ t $$):**
$$
t = 2.5 \times \frac{50}{2} \
t = 2.5 \times 25 \
t = 62.5 \text{ minutes}
$$

**Therefore:**
$$
\boxed{62.5 \text{ minutes}}
$$

---

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

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