A sphere, a meterstick, and the sun...

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Hello all! I think I'm on the right track with this problem, but it's always good to have someone double check, so here I am! Thanks for any corrections, etc that you are able to provide and for taking the time to read this post!

Problem:

"There is a large sphere on a horizontal field, on a sunny day, and at a certain time the shadow of the sphere reaches out a distance of 10m from the point where the sphere touches the ground. At the same time, a meter stick held vertically with one end on the ground casts a shadow of length two meters. What is the radius of the sphere in meters assuming the suns rays are parallel and the meter stick is a line segment?"

I first drew two triangles, one from the top of the sphere to the ground, then out 10m, and then connected by the hypotenuse... the second triangle is from the top of the meter stick to the ground, and out 2m. Then I set up a proportion, where 1/2=x/10 where x is the unknown diameter of the sphere. solving for x, I found x to be equal to five, and since that is the diameter, I divided by two to find 2.5m for the radius... Is this correct, or has my train completely fallen off the track? Thanks!

Arcainine
 
There is a large sphere on a horizontal field, on a sunny day, and at a certain time the shadow of the sphere reaches out a distance of 10m from the point where the sphere touches the ground. At the same time, a meter stick held vertically with one end on the ground casts a shadow of length two meters. What is the radius of the sphere in meters assuming the suns rays are parallel and the meter stick is a line segment?"

The shadow line does not touch the top center of the sphere.
1--Draw a circle with center C representing your sphere.
2--Draw a horizontal line beneath the sphere representing the ground.
3--From some distance to the left of the sphere at point A, draw a line from the ground tangent to the sphere at point P.
4--Locate the meter stick 2m to the right of point A.
5--Draw the vertical diameter of the sphere from point B on the ground and extend it beyond the sphere to intersect the extended ray AP at point D.
6--Triangles ABD, and CPD are similar, the two legs being in the ratio of 2/1.
7--The height of the meter stick is one half the shadow length.
8--Distance BD is one half 10 meters or 5 meters.
9--Lettting PC = r, the radius of the sphere, we can write (AP)^2 + (PC)^2 = (BD - r)^2 or (r/2)^2 + r^2 = (5 -- r)^2.
10-- Solve for r.
 
TchrWill said:
There is a large sphere on a horizontal field, on a sunny day, and at a certain time the shadow of the sphere reaches out a distance of 10m from the point where the sphere touches the ground. At the same time, a meter stick held vertically with one end on the ground casts a shadow of length two meters. What is the radius of the sphere in meters assuming the suns rays are parallel and the meter stick is a line segment?"

The shadow line does not touch the top center of the sphere.
1--Draw a circle with center C representing your sphere.
2--Draw a horizontal line beneath the sphere representing the ground.
3--From some distance to the left of the sphere at point A, draw a line from the ground tangent to the sphere at point P.
4--Locate the meter stick 2m to the right of point A.
5--Draw the vertical diameter of the sphere from point B on the ground and extend it beyond the sphere to intersect the extended ray AP at point D.
6--Triangles ABD, and CPD are similar, the two legs being in the ratio of 2/1.
7--The height of the meter stick is one half the shadow length.
8--Distance BD is one half 10 meters or 5 meters.
9--Lettting PC = r, the radius of the sphere, we can write (AP)^2 + (PC)^2 = (BD - r)^2 or (r/2)^2 + r^2 = (5 -- r)^2.
10-- Solve for r.
Click to expand...

Confused? Why wouldn't the shadow line touch the top of the circle? And, since it's a circle, wouldn't any tangential line to the surface of it be equidistant from the radius? I think maybe I'm missing something... Thanks!
 
ArcainineFalls531 wrote:

Confused? Why wouldn't the shadow line touch the top of the circle? And, since it's a circle, wouldn't any tangential line to the surface of it be equidistant from the radius? I think maybe I'm missing something... Thanks!

The shadow line is tangent to the circle 8.726º (arctan(1/2)from the top centerline point of the sphere. There is only one line tangent to the top centerline point of the circle which is parallel to the ground and cannot produce a shadow with a height to length ratio of 1/2.

Draw the picture and it will become obvious to you.
 
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