EXERCTSE 7 You may use a calculator in this exercise. Assume that the rates remain constant. (a) What is the speed, in km/h, of a car that travels a distance of 240 km in 3 hours? (b) The temperature in a room changed from $$ 9{ }^{\circ} \mathrm{C} $$ to $$ 27^{\circ} \mathrm{C} $$ in 3 minutes. What is the rate of change of temperature in $$ { }^{\circ} \mathrm{C} / $$ min? (c) If $$ 50 / $$ of liquid has a mass of 100 kg , what is the density of the liquid in $$ \mathrm{g} / \mathrm{cm}^{\prime} $$ ? Note: $$ 1 /=1000 \mathrm{~cm}^{3} $$ (d) If Aidan runs 10 km in 25 minutes, what is his speed in $$ \mathrm{km} / \mathrm{h} $$ ? (e) A container is filled, from empty, with water, at a rate of 30 lmin . (1) How much water will be in the container after an hour? (2) If the container has a capacity of 6 kl , how long does it take to fill the container up (f) A computer downloads data at 500 kilobytes per second. How many kilobytes can you download in 5 minutes? (g) A wheel turns at 900 rpm (revolutions per minute). (1) How many revolutions does the wheel make in 15 minutes? (2) How many revolutions does it make in 2 hours? (3) How long does it take for the wheel to make 45000 revolutions?

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Sure, let's work through each part of Exercise 7 step by step.

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### **(a) Speed of the Car**

**Problem:**
What is the speed, in km/h, of a car that travels a distance of 240 km in 3 hours?

**Solution:**
Speed is calculated using the formula:
$$
\text{Speed} = \frac{\text{Distance}}{\text{Time}}
$$
$$
\text{Speed} = \frac{240 \text{ km}}{3 \text{ h}} = 80 \text{ km/h}
$$

**Answer:**  
The speed of the car is **80 km/h**.

---

### **(b) Rate of Change of Temperature**

**Problem:**
The temperature in a room changed from $$9^{\circ}\mathrm{C}$$ to $$27^{\circ}\mathrm{C}$$ in 3 minutes. What is the rate of change of temperature in $$^{\circ}\mathrm{C}/$$min?

**Solution:**
Rate of change is calculated as:
$$
\text{Rate} = \frac{\Delta \text{Temperature}}{\Delta \text{Time}} = \frac{27^{\circ}\mathrm{C} - 9^{\circ}\mathrm{C}}{3 \text{ min}} = \frac{18^{\circ}\mathrm{C}}{3 \text{ min}} = 6^{\circ}\mathrm{C}/\text{min}
$$

**Answer:**  
The rate of change of temperature is **6 °C per minute**.

---

### **(c) Density of the Liquid**

**Problem:**
If 50 L of liquid has a mass of 100 kg, what is the density of the liquid in $$ \mathrm{g}/\mathrm{cm}^3 $$?
*Note: $$1 \text{ L} = 1000 \text{ cm}^3$$*

**Solution:**
First, convert mass to grams and volume to cubic centimeters:
$$
100 \text{ kg} = 100,000 \text{ g}
$$
$$
50 \text{ L} = 50 \times 1000 \text{ cm}^3 = 50,000 \text{ cm}^3
$$
Now, calculate density:
$$
\text{Density} = \frac{\text{Mass}}{\text{Volume}} = \frac{100,000 \text{ g}}{50,000 \text{ cm}^3} = 2 \text{ g/cm}^3
$$

**Answer:**  
The density of the liquid is **2 g/cm³**.

---

### **(d) Aidan’s Running Speed**

**Problem:**
If Aidan runs 10 km in 25 minutes, what is his speed in km/h?

**Solution:**
First, convert 25 minutes to hours:
$$
25 \text{ minutes} = \frac{25}{60} \text{ hours} = \frac{5}{12} \text{ hours}
$$
Now, calculate speed:
$$
\text{Speed} = \frac{10 \text{ km}}{\frac{5}{12} \text{ hours}} = 10 \times \frac{12}{5} = 24 \text{ km/h}
$$

**Answer:**  
Aidan’s speed is **24 km/h**.

---

### **(e) Filling the Container with Water**

**Problem:**
A container is filled, from empty, with water, at a rate of 30 L/min.
1. How much water will be in the container after an hour?
2. If the container has a capacity of 6 kL, how long does it take to fill the container up?

**Solution:**

**Part (1):**
$$
\text{Volume} = \text{Rate} \times \text{Time} = 30 \text{ L/min} \times 60 \text{ min} = 1800 \text{ L}
$$

**Part (2):**
Convert capacity to liters:
$$
6 \text{ kL} = 6 \times 1000 \text{ L} = 6000 \text{ L}
$$
Now, calculate time:
$$
\text{Time} = \frac{\text{Volume}}{\text{Rate}} = \frac{6000 \text{ L}}{30 \text{ L/min}} = 200 \text{ minutes}
$$
Alternatively, in hours and minutes:
$$
200 \text{ minutes} = 3 \text{ hours and } 20 \text{ minutes}
$$

**Answers:**
1. **1800 liters** of water after an hour.
2. **200 minutes** (or **3 hours and 20 minutes**) to fill the container.

---

### **(f) Data Download**

**Problem:**
A computer downloads data at 500 kilobytes per second. How many kilobytes can you download in 5 minutes?

**Solution:**
First, convert 5 minutes to seconds:
$$
5 \text{ minutes} = 5 \times 60 \text{ seconds} = 300 \text{ seconds}
$$
Now, calculate total data downloaded:
$$
\text{Total Data} = \text{Rate} \times \text{Time} = 500 \text{ KB/s} \times 300 \text{ s} = 150,000 \text{ KB}
$$

**Answer:**  
You can download **150,000 kilobytes** in 5 minutes.

---

### **(g) Wheel Revolutions**

**Problem:**
A wheel turns at 900 rpm (revolutions per minute).
1. How many revolutions does the wheel make in 15 minutes?
2. How many revolutions does it make in 2 hours?
3. How long does it take for the wheel to make 45,000 revolutions?

**Solution:**

**Part (1):**
$$
\text{Revolutions} = 900 \text{ rpm} \times 15 \text{ minutes} = 13,500 \text{ revolutions}
$$

**Part (2):**
Convert 2 hours to minutes:
$$
2 \text{ hours} = 2 \times 60 \text{ minutes} = 120 \text{ minutes}
$$
Now, calculate revolutions:
$$
\text{Revolutions} = 900 \text{ rpm} \times 120 \text{ minutes} = 108,000 \text{ revolutions}
$$

**Part (3):**
Find time required for 45,000 revolutions:
$$
\text{Time} = \frac{\text{Revolutions}}{\text{Rate}} = \frac{45,000 \text{ revolutions}}{900 \text{ rpm}} = 50 \text{ minutes}
$$

**Answers:**
1. **13,500 revolutions** in 15 minutes.
2. **108,000 revolutions** in 2 hours.
3. It takes **50 minutes** to make 45,000 revolutions.

---

Feel free to reach out if you have any further questions or need additional explanations!
**Exercise 7 Solutions:** **(a) Car Speed** - **Speed:** 80 km/h **(b) Temperature Change Rate** - **Rate:** 6 °C per minute **(c) Liquid Density** - **Density:** 2 g/cm³ **(d) Aidan’s Running Speed** - **Speed:** 24 km/h **(e) Filling the Container** 1. **After 1 Hour:** 1800 liters 2. **To Fill 6 kL:** 200 minutes (3 hours and 20 minutes) **(f) Data Download** - **5 Minutes:** 150,000 kilobytes **(g) Wheel Revolutions** 1. **15 Minutes:** 13,500 revolutions 2. **2 Hours:** 108,000 revolutions 3. **45,000 Revolutions:** 50 minutes

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