The menual of a car instructs the owner to inflate the tyres to pressure of 200 kPa. (a) What is the recommended gauge pressure ? (b) What is the recommended absolute pressure ? (c ) If after the required inflation of the tyres, the car is driven to a mountain peak. where the atmospheric pressrre is 10 % below that at sea level, what will be the tyre gauge read ? Atmospehric pressure ` =1.01 xx 10^5Pa.`

Explanation

To solve the problem step by step, we will address each part of the question systematically.

### Given Data:
- Recommended tire pressure = 200 kPa
- Atmospheric pressure at sea level = 1.01 × 10^5 Pa = 101 kPa
- Atmospheric pressure at mountain peak = 10% below sea level pressure

### (a) Recommended Gauge Pressure
The gauge pressure is the pressure relative to atmospheric pressure. Since the manual recommends inflating the tires to 200 kPa, the gauge pressure is simply:

**Gauge Pressure (P_g) = 200 kPa**

### (b) Recommended Absolute Pressure
The absolute pressure is the sum of the gauge pressure and the atmospheric pressure. We can express this mathematically as:

\[ P_{abs} = P_g + P_a \]

Where:
- \( P_{abs} \) = Absolute Pressure
- \( P_g \) = Gauge Pressure = 200 kPa
- \( P_a \) = Atmospheric Pressure = 101 kPa

Substituting the values:

\[ P_{abs} = 200 \, \text{kPa} + 101 \, \text{kPa} = 301 \, \text{kPa} \]

### (c) Gauge Pressure at Mountain Peak
First, we need to calculate the atmospheric pressure at the mountain peak, which is 10% less than the sea level pressure:

\[ P_a = 101 \, \text{kPa} - (10\% \, \text{of} \, 101 \, \text{kPa}) \]
\[ P_a = 101 \, \text{kPa} - 10.1 \, \text{kPa} = 90.9 \, \text{kPa} \]

Now, we know that the absolute pressure does not change when the car is driven to the mountain peak. Therefore, we can use the previously calculated absolute pressure:

\[ P_{abs} = 301 \, \text{kPa} \]

Now, we can find the new gauge pressure at the mountain peak using the formula:

\[ P_g = P_{abs} - P_a \]

Substituting the values:

\[ P_g = 301 \, \text{kPa} - 90.9 \, \text{kPa} = 210.1 \, \text{kPa} \]

### Final Answers:
(a) Recommended Gauge Pressure = **200 kPa**
(b) Recommended Absolute Pressure = **301 kPa**
(c) Gauge Pressure at Mountain Peak = **210.1 kPa**