Distance from Palo Alto to Beijing. The surface of the earth is reasonably approximated as a sphere with radius R = 6367.5km. A location on the earth’s surface is traditionally given by its latitude θ and its longitude λ, which correspond to angular distance from the equator and prime meridian, respectively. The 3-D coordinates of the location are given by

The 3-D coordinates of the location of a point on the earth's surface is given by (R₀ₐ ×cosλ ×cosθ,  R₀ₐ×sinλ×cosθ,  R₀ₐ×sinλ)  where λ =longitude and θ  = latitude

and the distance between points on the surface such as two cities  is given by P₁ = (x₁, y₁, z₁) and P₂ = (x₂, y₂, z₂) = |P₁ P₂| = √((x₂-x₁)² + (y₂-y₁)² + (z₂ - z₁)²)

Step-by-step explanation:

If the center of the earth is marked O the location of a point is given in longitude and latitude as λ   and  θ respectively then  where normally we have

-180 ° ≤ λ ≤ 180 ° and 90 ° ≤ θ ≤ 90°

However in cartesan coordinates, considering the radius of the earth and specifying the location of a point on the surface of the earth in three dimensions of the x, y and z coordinates with distance from the earth center = R where R = |Oa| = 6367.5km and  the center of the Earth is O,   then a point, a  on the surface of the earth has   the x, y and z Cartesian coordinates we have

a = (xₐ, yₐ, zₐ) = (R×cosλ×cosθ, R×sinλ×cosθ, R×sinλ)

Hence the 3-D coordinates of a location is given by

(R₀ₐ ×cosλ ×cosθ,  R₀ₐ×sinλ×cosθ,  R₀ₐ×sinλ) with R₀ₐ = the distance of the point a to the center of the earth

Distance between two points is given by P₁ = (x₁, y₁, z₁) and P₂ = (x₂, y₂, z₂) = |P₁ P₂| = √((x₂-x₁)² + (y₂-y₁)² + (z₂ - z₁)²)

answer

explanation

the area of a circle is calculated using the formula

or

we can use any of the above formulae to calculate the required. preferably, let us use the latter since the diameter is

we substitute the diameter into the formula to get:

this is the exact value for the area of the circle.

the correct answer is a