Two rhombuses have sides with lengths of #1 #. If one rhombus has a corner with an angle of #pi/12 # and the other has a corner with an angle of #(11pi)/12 #, what is the difference between the areas of the rhombuses?

Answer 1

Difference between the areas of the rhombuses is #0#.

As one rhombus has a corner with an angle of #pi/12#, the other angle of this rhombus is #pi-pi/12=(11pi)/12#.
The other rhombus has a corner with an angle of #(11pi)/12#, the other angle of this rhombus is #pi-(11pi)/12=pi/12#.
This means that their sides being #1#, as angles too are equal, areas of the rhombuses will also be equal and
the difference between the areas of the rhombuses is #0#.
Note: area of rhombus is given by #a^2sinA#, where #a# is its one side and #A# is an angle of rhombus. Observe that #sin(pi-A)=sinA#, hence it does not matter whether we select #A# as acute angle of rhombus or obtuse angle.
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Answer 2

The difference between the areas of the two rhombuses can be found by first determining their diagonals.

For a rhombus with side length ( s ) and an angle ( \theta ) between adjacent sides, the length of each diagonal is given by ( \frac{s}{2} \times \sqrt{4 + \sec^2(\theta)} ).

Using this formula:

  1. For the rhombus with an angle of ( \frac{\pi}{12} ): ( \text{Diagonal}_1 = \frac{1}{2} \times \sqrt{4 + \sec^2\left(\frac{\pi}{12}\right)} )

  2. For the rhombus with an angle of ( \frac{11\pi}{12} ): ( \text{Diagonal}_2 = \frac{1}{2} \times \sqrt{4 + \sec^2\left(\frac{11\pi}{12}\right)} )

Calculate these diagonal lengths and then use the area formula for a rhombus, which is ( \text{Area} = \frac{1}{2} \times \text{Diagonal}_1 \times \text{Diagonal}_2 ), to find the areas of the two rhombuses. Finally, subtract the smaller area from the larger area to get the difference.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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