# Two rhombuses have sides with lengths of #4 #. If one rhombus has a corner with an angle of #pi/12 # and the other has a corner with an angle of #(5pi)/12 #, what is the difference between the areas of the rhombuses?

Difference in Area

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The area of a rhombus can be calculated using the formula ( A = s^2 \sin(\theta) ), where ( s ) is the length of a side and ( \theta ) is the measure of one of the angles.

For the first rhombus with an angle of ( \pi/12 ): [ A_1 = 4^2 \sin\left(\frac{\pi}{12}\right) = 16\sin\left(\frac{\pi}{12}\right) ]

For the second rhombus with an angle of ( \frac{5\pi}{12} ): [ A_2 = 4^2 \sin\left(\frac{5\pi}{12}\right) = 16\sin\left(\frac{5\pi}{12}\right) ]

The difference between the areas of the two rhombuses is: [ \Delta A = A_2 - A_1 = 16\sin\left(\frac{5\pi}{12}\right) - 16\sin\left(\frac{\pi}{12}\right) ]

Using trigonometric identities or a calculator:

- ( \sin\left(\frac{\pi}{12}\right) \approx 0.2588 )
- ( \sin\left(\frac{5\pi}{12}\right) \approx 0.9659 )

Thus, [ \Delta A = 16(0.9659) - 16(0.2588) = 15.4144 - 4.1408 = 11.2736 ]

Therefore, the difference between the areas of the two rhombuses is approximately ( 11.27 ) square units.

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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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