Problem involving ratios to find the length of a shadow

[overhorizon]

Homework Statement

The shadow cast by a vertical pillar in Alexandria at noon during the summer solstice is found to be 1/8 the height of the pillar. The distance between Alexandria and Syene is 1/8 the Earth's radius. Is there a geometric connection between these two 1-to-8 ratios?

I already solved this problem. However, there is another part of the problem:

If Earth were smaller than it is, would the shadow of the vertical pillar in Alexandria have been longer or shorter at noon during the summer solstice?

Homework Equations

Just a ratio I used to solve the first part.

Pole shadow/Pole height = Alexandria and Syene distance / Earth radius

The Attempt at a Solution

I said the shadow would grow shorter. Since the Earth would be smaller, the distance between Alexandria and Syene would also be smaller. And due to it being a ratio, this would also effect the shadow.

However, I am really not certain of my answer and do not know how I could verify it or if I am correct.

 

[presbyope]

Homework Statement

The shadow cast by a vertical pillar in Alexandria at noon during the summer solstice is found to be 1/8 the height of the pillar. The distance between Alexandria and Syene is 1/8 the Earth's radius. Is there a geometric connection between these two 1-to-8 ratios?

I already solved this problem. However, there is another part of the problem:

If Earth were smaller than it is, would the shadow of the vertical pillar in Alexandria have been longer or shorter at noon during the summer solstice?

Homework Equations

Just a ratio I used to solve the first part.

Pole shadow/Pole height = Alexandria and Syene distance / Earth radius

The Attempt at a Solution

I said the shadow would grow shorter. Since the Earth would be smaller, the distance between Alexandria and Syene would also be smaller. And due to it being a ratio, this would also effect the shadow.

However, I am really not certain of my answer and do not know how I could verify it or if I am correct.

You forgot to tell us what Syene has to do with the problem. Is there a shadow there? But yes, on a smaller planet Alexandria would be closer to the equator and would have a smaller shadow.

EDIT: Oops I meant the tropic not the equator. Obviously Syene is on the tropic of cancer south of Alexandria. So there is no shadow there right?