Quito, a city in Ecuador and Kampala, a city situated in Uganda both lie on the Equator. The longitude of Quito is `82^(@)30'` W and that of Kampala is `37^(@)30'`E. What is the distance from Quito to Kampala.
(a) along the shortest surface path
(b) along a direct (through - the -Earth )path?(The radius of the Earth is `6.4xx10^(6)m`)

Answer

To solve the problem of finding the distance between Quito and Kampala, we need to calculate two types of distances:

(a) the shortest surface path along the Earth's surface, and
(b) the direct distance through the Earth.

### Step-by-Step Solution:

**Step 1: Understand the Longitudes**
- The longitude of Quito is \( 82^\circ 30' \) W, and the longitude of Kampala is \( 37^\circ 30' \) E.
- To find the total angle between the two cities, we need to add these two longitudes together:
\[ \text{Total angle} = 82^\circ 30' + 37^\circ 30' = 120^\circ \]

**Step 2: Convert the Angle to Radians**
- We need to convert degrees to radians for our calculations. The conversion factor is \( \frac{\pi \text{ radians}}{180^\circ} \).
- Therefore, the angle in radians is:
\[ \theta = 120^\circ \times \frac{\pi}{180} = \frac{2\pi}{3} \text{ radians} \]

**Step 3: Calculate the Shortest Surface Path (Arc Length)**
- The arc length \( L \) can be calculated using the formula:
\[ L = r \theta \]
where \( r \) is the radius of the Earth, given as \( 6.4 \times 10^6 \) m.
- Plugging in the values:
\[ L = 6.4 \times 10^6 \times \frac{2\pi}{3} \]
- Calculating this gives:
\[ L \approx 4.01 \times 10^5 \text{ m} \]

**Step 4: Calculate the Direct Distance Through the Earth**
- The direct distance can be calculated using the formula:
\[ d = 2r \sin\left(\frac{\theta}{2}\right) \]
- Since \( \theta = 120^\circ \), we have:
\[ \frac{\theta}{2} = 60^\circ \]
- Therefore, we can write:
\[ d = 2 \times 6.4 \times 10^6 \times \sin(60^\circ) \]
- Knowing that \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \):
\[ d = 2 \times 6.4 \times 10^6 \times \frac{\sqrt{3}}{2} \]
- This simplifies to:
\[ d = 6.4 \times 10^6 \times \sqrt{3} \]
- Calculating this gives:
\[ d \approx 1.1 \times 10^6 \text{ m} \]

### Final Answers:
(a) The shortest surface path from Quito to Kampala is approximately \( 4.01 \times 10^5 \text{ m} \).

(b) The direct distance through the Earth from Quito to Kampala is approximately \( 1.1 \times 10^6 \text{ m} \).