# Two rhombuses have sides with lengths of #1 #. If one rhombus has a corner with an angle of #(pi)/2 # and the other has a corner with an angle of #(3pi)/8 #, what is the difference between the areas of the rhombuses?

The difference between areas of two rhombuses is

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The area of a rhombus is given by the formula ( A = \frac{1}{2} \times d_1 \times d_2 ), where ( d_1 ) and ( d_2 ) are the lengths of its diagonals.

For the rhombus with a corner angle of ( \frac{\pi}{2} ), the diagonals are equal in length and each diagonal is ( \sqrt{2} ) (since it's a square). So, the area of this rhombus is ( A_1 = \frac{1}{2} \times \sqrt{2} \times \sqrt{2} = 1 ).

For the rhombus with a corner angle of ( \frac{3\pi}{8} ), the diagonals are not equal. We can use trigonometry to find the lengths of the diagonals. Let ( d_1 ) be the longer diagonal and ( d_2 ) be the shorter diagonal. Using the given angle, we can calculate ( d_1 ) and ( d_2 ) as ( d_1 = 1 ) and ( d_2 = 2\sin\left(\frac{3\pi}{16}\right) ).

Then, the area of this rhombus is ( A_2 = \frac{1}{2} \times 1 \times 2\sin\left(\frac{3\pi}{16}\right) ).

The difference in area between the two rhombuses is ( A_1 - A_2 ). Therefore, the difference between the areas is ( 1 - \sin\left(\frac{3\pi}{16}\right) ).

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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.

- Two opposite sides of a parallelogram each have a length of #2 #. If one corner of the parallelogram has an angle of #(2 pi)/3 # and the parallelogram's area is #15 #, how long are the other two sides?
- Two rhombuses have sides with lengths of #1 #. If one rhombus has a corner with an angle of #pi/6 # and the other has a corner with an angle of #(3pi)/4 #, what is the difference between the areas of the rhombuses?
- A parallelogram has sides with lengths of #16 # and #15 #. If the parallelogram's area is #48 #, what is the length of its longest diagonal?
- A parallelogram has sides A, B, C, and D. Sides A and B have a length of #3 # and sides C and D have a length of # 8 #. If the angle between sides A and C is #(3 pi)/4 #, what is the area of the parallelogram?
- A parallelogram has sides with lengths of #15 # and #12 #. If the parallelogram's area is #150 #, what is the length of its longest diagonal?

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