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Given latitude = 52^ N.

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3 parts
i

24690 km24690 \text{ km}

ii

14450 km14450 \text{ km}

iii

2751 km2751 \text{ km}

Step-by-step solution

11 steps

To solve this problem, we will use the Earth's radius R=6370 kmR = 6370 \text{ km} and π≈3.14159\pi \approx 3.14159.

Part i· 3 steps

**Length of the circle of latitude 52∘N52^\circ N

The formula for the length of a circle of latitude is L=2πRcos⁡ϕL = 2\pi R \cos \phi, where RR is the Earth's radius and ϕ\phi is the latitude.
Given latitude ϕ=52∘N\phi = 52^\circ N.

  1. Substitute the values into the formula.

    L=2×π×6370 km×cos⁡(52∘)L = 2 \times \pi \times 6370 \text{ km} \times \cos(52^\circ)

  2. Calculate the value.

    L=2×3.14159265×6370×0.61566147L = 2 \times 3.14159265 \times 6370 \times 0.61566147 L=24690.69 kmL = 24690.69 \text{ km}

  3. Round to a suitable number of significant figures (e.g., 4 significant figures).

    L≈24690 kmL \approx \boxed{24690 \text{ km}}

Part ii

14450 km14450 \text{ km}

**Distance BC along the longitude 76∘E76^\circ E

Points B and C are on the same longitude (76∘E76^\circ E).
Latitude of B is 52∘N52^\circ N.
Latitude of C is 78∘S78^\circ S.
Since the latitudes are in different hemispheres (North and South), the angular difference θ\theta is the sum of their absolute values.

  1. Calculate the angular difference θ\theta.

    θ=52∘+78∘=130∘\theta = 52^\circ + 78^\circ = 130^\circ

  2. The distance along a longitude (which is a great circle) is given by D=θ360∘×2πRD = \frac{\theta}{360^\circ} \times 2\pi R.

    Substitute the values into the formula.

    DBC=130∘360∘×2×π×6370 kmD_{BC} = \frac{130^\circ}{360^\circ} \times 2 \times \pi \times 6370 \text{ km}

  3. Calculate the value.

    DBC=1336×2×3.14159265×6370D_{BC} = \frac{13}{36} \times 2 \times 3.14159265 \times 6370 DBC=0.36111111×40023.8946D_{BC} = 0.36111111 \times 40023.8946 DBC=14453.70 kmD_{BC} = 14453.70 \text{ km}

  4. Round to a suitable number of significant figures (e.g., 4 significant figures).

    DBC≈14450 kmD_{BC} \approx \boxed{14450 \text{ km}}

Part iii

2751 km2751 \text{ km}

**Distance DC along the circle of latitude 78∘S78^\circ S

Points D and C are on the same circle of latitude (78∘S78^\circ S).
Longitude of D is 43∘W43^\circ W.
Longitude of C is 76∘E76^\circ E.
Since the longitudes are in different hemispheres (West and East), the angular difference α\alpha is the sum of their absolute values.

  1. Calculate the angular difference α\alpha.

    α=43∘+76∘=119∘\alpha = 43^\circ + 76^\circ = 119^\circ

  2. The distance along a circle of latitude is given by D=α360∘×2πRcos⁡ϕD = \frac{\alpha}{360^\circ} \times 2\pi R \cos \phi.

    Given latitude ϕ=78∘S\phi = 78^\circ S.
    Substitute the values into the formula.

    DDC=119∘360∘×2×π×6370 km×cos⁡(78∘)D_{DC} = \frac{119^\circ}{360^\circ} \times 2 \times \pi \times 6370 \text{ km} \times \cos(78^\circ)

  3. Calculate the value.

    DDC=119360×2×3.14159265×6370×0.20791169D_{DC} = \frac{119}{360} \times 2 \times 3.14159265 \times 6370 \times 0.20791169 DDC=0.33055555×40023.8946×0.20791169D_{DC} = 0.33055555 \times 40023.8946 \times 0.20791169 DDC=0.33055555×8321.030D_{DC} = 0.33055555 \times 8321.030 DDC=2751.00 kmD_{DC} = 2751.00 \text{ km}

  4. Round to a suitable number of significant figures (e.g., 4 significant figures).

    DDC≈2751 kmD_{DC} \approx \boxed{2751 \text{ km}}
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