I'm guessing the sum is supposed to be
Split the summand into partial fractions:
If , then
If , then
This means
Consider the th partial sum of the series:
The sum telescopes so that
and as , the second term vanishes and leaves us with
Distances in space are measured in light-years. The distance from Earth to a star is 6.8 × 1013 kilometers. What is the distance
2 answers:
I light year = 9.46 x 10¹² km
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Find the number of light years
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Answer: The distance from Earth to the star is 7.18 light-years.
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Find the number of light years
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Answer: The distance from Earth to the star is 7.18 light-years.
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Since the given hexagon is a regular hexagon all it's sides will be of equal length. Now, we know that the Area of any regular hexagon is given by:
Where is the area of the regular hexagon
is the side length of the regular hexagon
Also, it's Perimeter is given by:
Thus, all that we need to do is to find the side length of any one of the sides and to do that let us have a look at at the data of vertices points given and find out which points are definitely adjacent to each other and are also easy to calculate.
A quick search will yield that D(8, 0) and E(4, 0) are definitely adjacent to each other.
Please check the attached file here for a better understanding of the diagram of the original regular hexagon. Points D and E indeed are adjacent to each other.
Let us now find the distance between the points D and E using the distance formula which is as:
Where is the distance.
and
are the coordinates of points D and E respectively. (please note that interchanging the values of the coordinates will not alter the distance
)
Applying the above formula we get:
We know that this distance is the side length of the given regular hexagon.
Now, if the sides of the given regular polygon are reduced by 40%, then the new length of the sides will be:
Thus, the area of the smaller hexagon will be:
unit squared
and the new smaller perimeter will be:
unit
Which are the required answers.
