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

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To find the areas of the rhombuses, we use the formula for the area of a rhombus: A = (1/2) * d1 * d2, where d1 and d2 are the diagonals of the rhombus.

Given that the sides of each rhombus are 8 units long, we can find the lengths of the diagonals using trigonometry.

For the first rhombus with an angle of π/12: The diagonal can be found using the law of sines: sin(π/12) = (d1/8) Solving for d1, we get: d1 = 8 * sin(π/12)

Similarly, since the opposite angle is also π/12, the other diagonal is also 8 * sin(π/12).

For the second rhombus with an angle of (3π)/8: Using the same method: sin((3π)/8) = (d2/8) Solving for d2, we get: d2 = 8 * sin((3π)/8)

To find the areas of the rhombuses, we use the formula mentioned earlier: Area1 = (1/2) * (8 * sin(π/12)) * (8 * sin(π/12)) Area2 = (1/2) * (8 * sin((3π)/8)) * (8 * sin((3π)/8))

Then, we find the difference between the areas: Difference = Area2 - Area1

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To find the areas of the rhombuses, we use the formula:

[ \text{Area} = \frac{d_1 \times d_2}{2} ]

Where ( d_1 ) and ( d_2 ) are the lengths of the diagonals of the rhombus.

Given that the sides of the rhombuses are ( 8 ) units each, we can find the lengths of the diagonals using trigonometry.

For the rhombus with an angle of ( \frac{\pi}{12} ):

Using trigonometry, we can find ( d_1 ) and ( d_2 ) as follows:

[ d_1 = 2 \times 8 \times \sin\left(\frac{\pi}{12}\right) ] [ d_2 = 2 \times 8 \times \sin\left(\frac{\pi}{2} - \frac{\pi}{12}\right) ]

For the rhombus with an angle of ( \frac{3\pi}{8} ):

[ d_1' = 2 \times 8 \times \sin\left(\frac{3\pi}{8}\right) ] [ d_2' = 2 \times 8 \times \sin\left(\frac{\pi}{2} - \frac{3\pi}{8}\right) ]

Now, calculate the areas of each rhombus using the formula:

[ \text{Area}_1 = \frac{d_1 \times d_2}{2} ] [ \text{Area}_2 = \frac{d_1' \times d_2'}{2} ]

Finally, find the difference between the areas:

[ \text{Difference} = |\text{Area}_1 - \text{Area}_2| ]

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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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