Question

5. Asteroid: Anna and Beth landed on a huge asteroid. To measure its size, they walked 7 miles away from the spaceship. Anna waked towards the sun and Beth away from sun. Anna observed that she cast a shadow that made 40° with the vertical, while Beth observed her shadow at 50º. Both shadows are cast in the same direction. What is the circumference of the asteroid? What is its radius? RAYS SHADOWS

Answer 1

ن - ماز - - -

Ray R1 falls on Beth BB', casting shadow from B' to C. BC makes 50° with the vertical. Ray R2 falls on Anna AA', the ray makes 40° with the vertical. Both Anna and Beth are standing normal to the surface of asteroid. Assuming asteroid to be a sphere, extended lines AA' and BB' meets at the centre of the astroid O, let the angle between the two lines is θ.

To calculate θ we construct line R3 which is parallel to sunrays and passes through the center. Let x be the angle subtended by AO on Ray R3.

Since R2 is parallel to R3

Lx = 40° (alternate angles)

Since R1 is parallel to R3

20 + Zx = 50° (alternate angles)

20 + 40 = 50°

Zo = 10°

In radians

Zo= 10° x 180°

poi

The distance between Anna and Beth is 7miles, arc A'B'=7miles

We know that

drc radius

dlc radius =

RE 7 miles 7/18

R = 40.1 miles

The circumference is

C= 2TR

C = 27 x 40.1 miles)

C = 252 miles