Two rhombuses have sides with lengths of #4 #. 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?

Answer 1

Difference in areas between the two rhombuses is 3.3136

Area of rhombus #= (1/2) * d_1 * d_2 or a * h#
Where #d_1 , d_2 # are the diagonals, a is the side and h is the altitude.

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

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Answer 2

To find the difference between the areas of the two rhombuses, use the formula for the area of a rhombus, which is given by A = d1 * d2 / 2, where d1 and d2 are the lengths of the diagonals.

For a rhombus with a corner angle of π/6, the diagonals can be found using trigonometry:

  • The length of one diagonal is 4 / sin(π/6) = 8.
  • The length of the other diagonal is 4 / sin(π - π/6) = 4 / sin(5π/6) = 8.

Thus, for the rhombus with the corner angle of π/6, both diagonals have lengths of 8.

For a rhombus with a corner angle of 3π/4, the diagonals can also be found:

  • The length of one diagonal is 4 / sin(3π/4) = 4 / (sqrt(2)/2) = 8 * sqrt(2).
  • The length of the other diagonal is 4 / sin(π - 3π/4) = 4 / (sqrt(2)/2) = 8 * sqrt(2).

Thus, for the rhombus with the corner angle of 3π/4, both diagonals have lengths of 8 * sqrt(2).

Now, calculate the areas of the two rhombuses:

  • For the rhombus with the corner angle of π/6, the area is (8 * 8) / 2 = 32 square units.
  • For the rhombus with the corner angle of 3π/4, the area is (8 * 8 * sqrt(2)) / 2 = 32 * sqrt(2) square units.

The difference between the areas is 32 * sqrt(2) - 32 square units.

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Answer from HIX Tutor

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.

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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