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")#