# Two rhombuses have sides with lengths of #8 #. 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?

Difference in areas between the two rhombuses is 13.2552

Area of rhombus

Where

In this case we will use the formula Area = a * h.

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The area of a rhombus can be calculated using the formula: ( A = \frac{1}{2} \times d_1 \times d_2 ), where ( d_1 ) and ( d_2 ) are the lengths of the diagonals.

For the rhombus with an angle of ( \frac{\pi}{6} ), we can calculate the area using trigonometry to find the diagonals: ( d_1 = 2 \times 8 \times \sin\left(\frac{\pi}{6}\right) ) and ( d_2 = 2 \times 8 \times \sin\left(\frac{5\pi}{6}\right) ).

For the rhombus with an angle of ( \frac{3\pi}{4} ), we can similarly calculate the diagonals: ( d_1 = 2 \times 8 \times \sin\left(\frac{3\pi}{4}\right) ) and ( d_2 = 2 \times 8 \times \sin\left(\frac{\pi}{4}\right) ).

Once we have the diagonals for both rhombuses, we can find their areas and then calculate the difference.

Let's proceed with the calculations.For the rhombus with an angle of ( \frac{\pi}{6} ), the diagonals are:

( d_1 = 2 \times 8 \times \sin\left(\frac{\pi}{6}\right) = 16 \times \frac{1}{2} = 8 )

( d_2 = 2 \times 8 \times \sin\left(\frac{5\pi}{6}\right) = 16 \times \frac{\sqrt{3}}{2} = 8\sqrt{3} )

Area of this rhombus: ( A_1 = \frac{1}{2} \times 8 \times 8\sqrt{3} = 32\sqrt{3} )

For the rhombus with an angle of ( \frac{3\pi}{4} ), the diagonals are:

( d_1 = 2 \times 8 \times \sin\left(\frac{3\pi}{4}\right) = 16 \times \frac{\sqrt{2}}{2} = 8\sqrt{2} )

( d_2 = 2 \times 8 \times \sin\left(\frac{\pi}{4}\right) = 16 \times \frac{\sqrt{2}}{2} = 8\sqrt{2} )

Area of this rhombus: ( A_2 = \frac{1}{2} \times 8\sqrt{2} \times 8\sqrt{2} = 64 )

The difference between the areas of the two rhombuses is:

( A_2 - A_1 = 64 - 32\sqrt{3} )

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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.

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