A person is standing at the edge of the water and looking out at the ocean. The height of the person's eyes above the water is h= 1.6 meters, and the radius of the earth is R= 6.38 *10^6 meters. (a) How far is it to the horizon? (b) Express (a) in miles.

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Answer 1 · 2018-01-10T07:24:45

(a) 4.5 km

(b) 2.8 miles

(a)

If R is the radius of the Earth and h is the height of the person then the diagram shows we have a right angle triangle of side R, R + h and d.

Applying Pythagoras we get:

$sf(d^2+R^2=(R+h)^2)$

$:.$ $sf(d^2+R^2=(R+h)(R+h))$

$sf(d^2+cancel(R^2)=cancel(R^2)+hR+hR+h^2)$

$sf(d^2=2hR+h^2)$

$sf(d^2=(2xx1.6xx6.38xx10^6)+1.6^2)$

$sf(d^2=20.416xx10^6+2.56)$

Because $sf(20.416xx10^6)$ is much greater than $sf(2.56)$ we can say that $sf((20.416xx10^6+2.56)rArr20.416xx10^6)$

$:.$ $sf(d^2=20.416xx10^6)$

$sf(d=sqrt(20.416xx10^6)=4.5xx10^3color(white)(x)m)$

$sf(d=4.5color(white)(x)"km")$

(b)

$sf(d=4.5xx5/8=2.8color(white)(x)"miles")$