Ecotourists use their global positioning system indicator to determine their loc

Question

Ecotourists use their global positioning system indicator to determine their location inside a botanical garden as latitude 0.002 20 degree south of the equator, longitude 75.640 35 degrees west. They wish to visit a tree at latitude 0.001 59 degree north, longitude 75.644 55 degrees west. (a) Determine the straight-line distance and the direction in which they can walk to reach the tree as follows. First model the Earth as a sphere of radius 6.37 Mm to determine the westward and northward displacement components required, in meters. displacement west __m displacement north __m Now model the Earth as a flat surface to complete the calculation. magnitude __m direction __

Explanation / Answer

Firstly, both locations are so close to the equator, that it is of no extra significance to use the exact formula (for a spherical earth), which is : (Radius of latitude circle) = (Radius of Earth) * cos(latitude) By not using it, the circumference of both latitude circles is only about 0.1 metre, or less, different from the circumference of the equator. So for such a small difference between the longitude locations, the distance will be accurate to much less than 0.1 metre. For all intents and purposes then, distances around the latitude circles may be taken as equivalent to distances around the equator. The westward displacement is 75.64495º - 75.64300º = 0.00195º. If radius of Earth is 6370000 m, then circumference is 2p*6370000 m. But 2p*6370000 m is equivalent to 360º. Therefore, 0.00195º is equivalent to 2p*6370000 * (0.00195/360) m = 217 m. The northward displacement is 0.00405º + 0.00134º = 0.00539º. Again, the circumference of a great circle (this time through the tree and both the north and south poles as well) is 2p*6370000 m, equivalent to 360º. Therefore, 0.00539º is equivalent to 2p*6370000 * (0.00539/360) m = 599 m. Modelling the earth as a flat surface, we have a right-angled triangle with legs equal to 217 m and 599 m, so by Pythagoras, the hypotenuse is : v(217^2 + 599^2) = 637 m If A is the angle of the triangle at the garden, then : tan(A) = 599/217 Therefore, A = arctan(599/217) = 70º. Thus, magnitude = 637 m and direction = 70º north of west

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