# Two rhombuses have sides with lengths of #8 #. If one rhombus has a corner with an angle of #(5pi)/12 # and the other has a corner with an angle of #(3pi)/8 #, what is the difference between the areas of the rhombuses?

One rhombus has larger area than the other by

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

For the first rhombus, if one angle is ( \frac{5\pi}{12} ), then the adjacent angle is also ( \frac{5\pi}{12} ), making the other two angles also ( \frac{5\pi}{12} ). The diagonals of the rhombus bisect each other at right angles, forming four right-angled triangles. Using trigonometric functions, we can find that the lengths of the diagonals are ( 8\sin(\frac{5\pi}{12}) ) and ( 8\cos(\frac{5\pi}{12}) ).

For the second rhombus, if one angle is ( \frac{3\pi}{8} ), then the adjacent angle is also ( \frac{3\pi}{8} ), making the other two angles also ( \frac{3\pi}{8} ). Using similar reasoning as before, the lengths of the diagonals are ( 8\sin(\frac{3\pi}{8}) ) and ( 8\cos(\frac{3\pi}{8}) ).

Now, we can calculate the areas of both rhombuses:

For the first rhombus: [ \text{Area}_1 = \frac{8\sin(\frac{5\pi}{12}) \times 8\cos(\frac{5\pi}{12})}{2} ]

For the second rhombus: [ \text{Area}_2 = \frac{8\sin(\frac{3\pi}{8}) \times 8\cos(\frac{3\pi}{8})}{2} ]

Subtracting the areas gives us the difference: [ \text{Difference} = \text{Area}_1 - \text{Area}_2 ]

[ \text{Difference} = \left(\frac{8\sin(\frac{5\pi}{12}) \times 8\cos(\frac{5\pi}{12})}{2}\right) - \left(\frac{8\sin(\frac{3\pi}{8}) \times 8\cos(\frac{3\pi}{8})}{2}\right) ]

[ \text{Difference} = 32\left(\sin(\frac{5\pi}{12})\cos(\frac{5\pi}{12}) - \sin(\frac{3\pi}{8})\cos(\frac{3\pi}{8})\right) ]

[ \text{Difference} \approx 32 \times (0.5176 - 0.3827) ]

[ \text{Difference} \approx 32 \times 0.1349 ]

[ \text{Difference} \approx 4.3168 ]

Therefore, the difference between the areas of the two rhombuses is approximately ( 4.3168 ).

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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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