Answer :
Sure, let's analyze the problem step by step and find the time difference between Johannesburg and the specified longitudes:
Given:
- Johannesburg's longitude: [tex]\(28.0473^{\circ}E\)[/tex]
- The longitudes to compare:
- [tex]\(120^{\circ}W\)[/tex]
- [tex]\(135^{\circ}E\)[/tex]
- [tex]\(60^{\circ}E\)[/tex]
- [tex]\(75^{\circ}W\)[/tex]
- [tex]\(105^{\circ}E\)[/tex]
Step-by-Step Solution:
1. Calculate the time difference:
The time difference between two longitudes can be calculated by using the formula:
[tex]\[ \text{Time difference (in hours)} = \frac{\text{Longitude difference}}{15} \][/tex]
2. Longitude differences and calculation:
- For [tex]\(120^{\circ}W\)[/tex]:
[tex]\[ \text{Longitude difference} = 28.0473^{\circ}E - (-120^{\circ}W) = 28.0473 + 120 = 148.0473^{\circ} \][/tex]
[tex]\[ \text{Time difference} = \frac{148.0473}{15} \approx 9.8698 \text{ hours} \][/tex]
Since Johannesburg is ahead, the local time in Johannesburg is approximately 9.87 hours ahead of [tex]\(120^{\circ}W\)[/tex].
- For [tex]\(135^{\circ}E\)[/tex]:
[tex]\[ \text{Longitude difference} = 135^{\circ}E - 28.0473^{\circ}E = 135 - 28.0473 = 106.9527^{\circ} \][/tex]
[tex]\[ \text{Time difference} = \frac{106.9527}{15} \approx 7.13018 \text{ hours} \][/tex]
Johannesburg is approximately 7.13 hours behind [tex]\(135^{\circ}E\)[/tex].
- For [tex]\(60^{\circ}E\)[/tex]:
[tex]\[ \text{Longitude difference} = 60^{\circ}E - 28.0473^{\circ}E = 60 - 28.0473 = 31.9527^{\circ} \][/tex]
[tex]\[ \text{Time difference} = \frac{31.9527}{15} \approx 2.13018 \text{ hours} \][/tex]
Johannesburg is approximately 2.13 hours behind [tex]\(60^{\circ}E\)[/tex].
- For [tex]\(75^{\circ}W\)[/tex]:
[tex]\[ \text{Longitude difference} = 28.0473^{\circ}E - (-75^{\circ}W) = 28.0473 + 75 = 103.0473^{\circ} \][/tex]
[tex]\[ \text{Time difference} = \frac{103.0473}{15} \approx 6.8698 \text{ hours} \][/tex]
Since Johannesburg is ahead, the local time in Johannesburg is approximately 6.87 hours ahead of [tex]\(75^{\circ}W\)[/tex].
- For [tex]\(105^{\circ}E\)[/tex]:
[tex]\[ \text{Longitude difference} = 105^{\circ}E - 28.0473^{\circ}E = 105 - 28.0473 = 76.9527^{\circ} \][/tex]
[tex]\[ \text{Time difference} = \frac{76.9527}{15} \approx 5.130179 \text{ hours} \][/tex]
Johannesburg is approximately 5.13 hours behind [tex]\(105^{\circ}E\)[/tex].
Summary of time differences:
[tex]\[ \begin{align*} 120^{\circ}W & : \text{Approx. 9.87 hours behind Johannesburg} \\ 135^{\circ}E & : \text{Approx. 7.13 hours ahead of Johannesburg} \\ 60^{\circ}E & : \text{Approx. 2.13 hours ahead of Johannesburg} \\ 75^{\circ}W & : \text{Approx. 6.87 hours behind Johannesburg} \\ 105^{\circ}E & : \text{Approx. 5.13 hours ahead of Johannesburg} \\ \end{align*} \][/tex]
Thus, the time differences are as follows:
- [tex]\(120^{\circ}W\)[/tex]: 9.87 hours behind Johannesburg
- [tex]\(135^{\circ}E\)[/tex]: 7.13 hours ahead of Johannesburg
- [tex]\(60^{\circ}E\)[/tex]: 2.13 hours ahead of Johannesburg
- [tex]\(75^{\circ}W\)[/tex]: 6.87 hours behind Johannesburg
- [tex]\(105^{\circ}E\)[/tex]: 5.13 hours ahead of Johannesburg
Given:
- Johannesburg's longitude: [tex]\(28.0473^{\circ}E\)[/tex]
- The longitudes to compare:
- [tex]\(120^{\circ}W\)[/tex]
- [tex]\(135^{\circ}E\)[/tex]
- [tex]\(60^{\circ}E\)[/tex]
- [tex]\(75^{\circ}W\)[/tex]
- [tex]\(105^{\circ}E\)[/tex]
Step-by-Step Solution:
1. Calculate the time difference:
The time difference between two longitudes can be calculated by using the formula:
[tex]\[ \text{Time difference (in hours)} = \frac{\text{Longitude difference}}{15} \][/tex]
2. Longitude differences and calculation:
- For [tex]\(120^{\circ}W\)[/tex]:
[tex]\[ \text{Longitude difference} = 28.0473^{\circ}E - (-120^{\circ}W) = 28.0473 + 120 = 148.0473^{\circ} \][/tex]
[tex]\[ \text{Time difference} = \frac{148.0473}{15} \approx 9.8698 \text{ hours} \][/tex]
Since Johannesburg is ahead, the local time in Johannesburg is approximately 9.87 hours ahead of [tex]\(120^{\circ}W\)[/tex].
- For [tex]\(135^{\circ}E\)[/tex]:
[tex]\[ \text{Longitude difference} = 135^{\circ}E - 28.0473^{\circ}E = 135 - 28.0473 = 106.9527^{\circ} \][/tex]
[tex]\[ \text{Time difference} = \frac{106.9527}{15} \approx 7.13018 \text{ hours} \][/tex]
Johannesburg is approximately 7.13 hours behind [tex]\(135^{\circ}E\)[/tex].
- For [tex]\(60^{\circ}E\)[/tex]:
[tex]\[ \text{Longitude difference} = 60^{\circ}E - 28.0473^{\circ}E = 60 - 28.0473 = 31.9527^{\circ} \][/tex]
[tex]\[ \text{Time difference} = \frac{31.9527}{15} \approx 2.13018 \text{ hours} \][/tex]
Johannesburg is approximately 2.13 hours behind [tex]\(60^{\circ}E\)[/tex].
- For [tex]\(75^{\circ}W\)[/tex]:
[tex]\[ \text{Longitude difference} = 28.0473^{\circ}E - (-75^{\circ}W) = 28.0473 + 75 = 103.0473^{\circ} \][/tex]
[tex]\[ \text{Time difference} = \frac{103.0473}{15} \approx 6.8698 \text{ hours} \][/tex]
Since Johannesburg is ahead, the local time in Johannesburg is approximately 6.87 hours ahead of [tex]\(75^{\circ}W\)[/tex].
- For [tex]\(105^{\circ}E\)[/tex]:
[tex]\[ \text{Longitude difference} = 105^{\circ}E - 28.0473^{\circ}E = 105 - 28.0473 = 76.9527^{\circ} \][/tex]
[tex]\[ \text{Time difference} = \frac{76.9527}{15} \approx 5.130179 \text{ hours} \][/tex]
Johannesburg is approximately 5.13 hours behind [tex]\(105^{\circ}E\)[/tex].
Summary of time differences:
[tex]\[ \begin{align*} 120^{\circ}W & : \text{Approx. 9.87 hours behind Johannesburg} \\ 135^{\circ}E & : \text{Approx. 7.13 hours ahead of Johannesburg} \\ 60^{\circ}E & : \text{Approx. 2.13 hours ahead of Johannesburg} \\ 75^{\circ}W & : \text{Approx. 6.87 hours behind Johannesburg} \\ 105^{\circ}E & : \text{Approx. 5.13 hours ahead of Johannesburg} \\ \end{align*} \][/tex]
Thus, the time differences are as follows:
- [tex]\(120^{\circ}W\)[/tex]: 9.87 hours behind Johannesburg
- [tex]\(135^{\circ}E\)[/tex]: 7.13 hours ahead of Johannesburg
- [tex]\(60^{\circ}E\)[/tex]: 2.13 hours ahead of Johannesburg
- [tex]\(75^{\circ}W\)[/tex]: 6.87 hours behind Johannesburg
- [tex]\(105^{\circ}E\)[/tex]: 5.13 hours ahead of Johannesburg
