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An geometry maths question.


mkerala
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An ant lives on the surface of a cube with edges of length 7cm. It is currentlylocated on an edge x cm from one of its ends. While traveling on the surface of the cube,it has to reach the grain located on the opposite edge (also at a distance xcm from oneof its ends) as shown below.

(i) What is the length of the shortest route to the grain if x = 2cm? How many routes ofthis length are there?

(ii) Find an x for which there are four distinct shortest length routes to the grain

image.png

Please tell the steps you have followed to arrive at the solution.

Edited by mkerala
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What would be easier to work with than a cube?

 

A sphere, in whch case the geodesics are known to every school child -- great circles.

 

An ant lives on the surface of a cube with edges of length 7cm. It is currentlylocated on an edge x cm from one of its ends. While traveling on the surface of the cube,it has to reach the grain located on the opposite edge (also at a distance xcm from oneof its ends) as shown below.

(i) What is the length of the shortest route to the grain if x = 2cm? How many routes ofthis length are there?

(ii) Find an x for which there are four distinct shortest length routes to the grain

image.png

Please tell the steps you have followed to arrive at the solution.

 

This is pretty clearly a class assignment, or should be.

 

You need to show your work in a reasosnable attempt to solve this problem. Then we will help you.

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A sphere, in whch case the geodesics are known to every school child -- great circles.

 

 

 

This is pretty clearly a class assignment, or should be.

 

You need to show your work in a reasosnable attempt to solve this problem. Then we will help you.

I don't want to give too much away, but I was thinking of a different orb society!

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An ant lives on the surface of a cube with edges of length 7cm. Please tell the steps you have followed to arrive at the solution.

 

You say the ant lives on the surface of the cube (not just on the edge), so the cube is solid. So, with each side being 7cm's, the answer has to be that the ant moves straight up one side and then straight across the top. That's 14 cm's. Unless I'm an idiot (which hasn't been proven in a court of law yet!).

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You say the ant lives on the surface of the cube (not just on the edge), so the cube is solid. So, with each side being 7cm's, the answer has to be that the ant moves straight up one side and then straight across the top. That's 14 cm's. Unless I'm an idiot (which hasn't been proven in a court of law yet!).

That's the obvious answer - unfortunately its wrong. What if x was 0cm? Give you any ideas?

Edited by TonyMcC
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You say the ant lives on the surface of the cube (not just on the edge), so the cube is solid. So, with each side being 7cm's, the answer has to be that the ant moves straight up one side and then straight across the top. That's 14 cm's. Unless I'm an idiot (which hasn't been proven in a court of law yet!).

 

Think about cutting the cube along a few edges so that it can be unfolded and laid flat. That will not change distances of lines and curbes on the cube. If you make the cuts cleverly the answer will become clear.

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Think about cutting the cube along a few edges so that it can be unfolded and laid flat. That will not change distances of lines and curbes on the cube. If you make the cuts cleverly the answer will become clear.

The different orb society = flat earth society lol.

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That's the obvious answer - unfortunately its wrong. What if x was 0cm? Give you any ideas?

 

OH! You're pretty smart in my book! I won't give it away. All I'll say is, thank god there's a square root button on my calculator, and thank god for Pythagoras.

post-54460-0-51992800-1329518965_thumb.jpg

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They have also given me an example

 

Questions on three dimensional geometry sometimes require the student to consider a

two-dimensional representation of the underlying object and use methods of plane

geometry to arrive at the solution. Here is one such example.

Example: An ant lives on the surface of a regular tetrahedron with edges of length 3cm.

It is currently at the midpoint of one of the edges and has to travel to the midpoint of the

opposite edge where a grain is located (see figure). What is the length (in cm) of the

shortest route to the destination assuming that the ant can only travel along the surface

of the tetrahedron?

Solution: The ant has several routes by which it can reach the grain. For instance, it can

travel to the vertex C and move along edge CD.

 

image1.png

 

 

The idea behind finding the shortest route is to embed the surface of the tetrahedron on

a plane. This is done by opening the tetrahedron along some edges and spreading it out.

For example, the figure on the right is a planar representation containing the triangular

faces ABC and ACD. Notice that ABCD is a rhombus of length 3cm and the segment

joining ant and grain (which is the shortest route) is parallel to the base and thus of

length 3cm as well.

Now use the same idea to solve the problem below where the tetrahedron is replaced by

a cube.

 

Please use the example above to solve the problem.

 

 

 

 

 

Edited by mkerala
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