Given point B at latitude 12° N and longitude 110° W, and point A is 8008 km due south of B. Calculate the position of A. Then calculate the shorter distance in km between A and C along the parallel o

Here's how to solve the problem:

a) Determine:

a) (i) The position of A.

Step 1: Calculate the difference in latitude ($\Delta\phi$) corresponding to the distance of $8008 \text{ km}$.
The distance along a meridian is given by $D = \frac{\Delta\phi}{360^\circ} \times 2\pi R$.
Given $D = 8008 \text{ km}$, $R = 6370 \text{ km}$, and $\pi = \frac{22}{7}$.

$$8008 = \frac{\Delta\phi}{360^\circ} \times 2 \times \frac{22}{7} \times 6370$$ $$8008 = \frac{\Delta\phi}{360^\circ} \times \frac{44}{7} \times 6370$$ $$8008 = \frac{\Delta\phi}{360^\circ} \times 44 \times 910$$ $$8008 = \frac{\Delta\phi}{360^\circ} \times 40040$$ $$\Delta\phi = \frac{8008 \times 360^\circ}{40040}$$ $$\Delta\phi = \frac{8008 \times 36^\circ}{4004}$$ $$\Delta\phi = 2 \times 36^\circ$$ $$\Delta\phi = 72^\circ$$

Step 2: Determine the latitude of A.
Point B is at $12^\circ \text{ N}$. Point A is $8008 \text{ km}$ south of B, so its latitude will be $12^\circ \text{ N} - 72^\circ$.

$$\text{Latitude of A} = 12^\circ - 72^\circ = -60^\circ$$

This means the latitude of A is $60^\circ \text{ S}$.
Since A is directly south of B, they share the same longitude.
The longitude of B is $110^\circ \text{ W}$.

Step 3: State the position of A.
The position of A is $\boxed{(60^\circ \text{ S}, 110^\circ \text{ W})}$.

a) (ii) The shorter distance in kilometres between A and C along the parallel of latitude.

Step 1: Identify the coordinates of A and C.
From (a)(i), A is $(60^\circ \text{ S}, 110^\circ \text{ W})$.
Point C is on longitude $160^\circ \text{ E}$ and on the same latitude as A.
So, C is $(60^\circ \text{ S}, 160^\circ \text{ E})$.

Step 2: Calculate the difference in longitude ($\Delta\lambda$) between A and C.
Since A is in the West and C is in the East, we add their longitudes:

$$\Delta\lambda = 110^\circ + 160^\circ = 270^\circ$$

For the shorter distance, we take the smaller angle around the Earth:

$$\text{Shorter } \Delta\lambda = 360^\circ - 270^\circ = 90^\circ$$

Step 3: Calculate the radius of the parallel of latitude ($r$).
The latitude is $\phi = 60^\circ$.

$$r = R \cos \phi = 6370 \cos 60^\circ$$ $$r = 6370 \times 0.5 = 3185 \text{ km}$$

Step 4: Calculate the distance between A and C along the parallel of latitude.
The distance $D_{AC} = \frac{\text{Shorter } \Delta\lambda}{360^\circ} \times 2\pi r$.

$$D_{AC} = \frac{90^\circ}{360^\circ} \times 2 \times \frac{22}{7} \times 3185$$ $$D_{AC} = \frac{1}{4} \times 2 \times \frac{22}{7} \times 3185$$ $$D_{AC} = \frac{1}{2} \times \frac{22}{7} \times 3185$$ $$D_{AC} = \frac{11}{7} \times 3185$$ $$D_{AC} = 11 \times 455$$ $$$$