Question

Во сколько раз увеличится объем правильного тетраэдра если все его ребра увеличить в 6 раз

Answer 1

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Answer 2

Volume Increase of a Regular Tetrahedron

To calculate the increase in volume of a regular tetrahedron when all its edges are increased by 6 times, we can use the fact that the volume of a regular tetrahedron is given by the formula:

V = (a^3) / (6√2)

Where: - V = Volume of the tetrahedron - a = Length of the edge

If we increase the length of each edge by 6 times, the new length of the edge will be 6a. Therefore, the new volume (V') will be:

V' = ((6a)^3) / (6√2)

Simplifying this, we get:

V' = (216a^3) / (6√2) V' = 36a^3 / √2

To find the increase in volume, we can divide V' by V:

Increase in volume = V' / V Increase in volume = (36a^3 / √2) / (a^3 / (6√2)) Increase in volume = 36 / 1 Increase in volume = 36

Therefore, the volume of the regular tetrahedron will increase by 36 times if all its edges are increased by 6 times.