A pyramid has a base in the shape of a rhombus and a peak directly above the base's center. The pyramid's height is #5 #, its base has sides of length #4 #, and its base has a corner with an angle of #(3 pi)/8 #. What is the pyramid's surface area?

Answer 1

Total Surface Area #T S A = A_b + L_s = 14.78 + 43.12 = 57.9# sq units

Base area of the pyramid #A_b = l^2 sin theta#

#A_b = 4^2 sin ((3pi)/8) = 14.78#

Slant height of the pyramid #s = sqrt((l/2)^2 + h^2) #

#s = sqrt((4/2)^2 + 5^2) = 5.39#

Lateral Surface Area of pyramid (#L_s#) = 4 * Area of slant triangle (#4 * A_s#)

#L_s = 4 * A_s = 4 * (1/2) * l * s = 4 * (1/2) * 4 * 5.39 = 43.12#

Total Surface Area #T S A = A_b + L_s = 14.78 + 43.12 = 57.9# sq units

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Answer 2

To find the surface area of the pyramid, you need to calculate the area of its base and the area of its four triangular faces.

  1. Area of the Base (Rhombus): The area of a rhombus can be calculated using the formula: ( \text{Area} = \frac{1}{2} \times \text{diagonal}_1 \times \text{diagonal}_2 ) Given that the diagonals of a rhombus bisect each other at right angles, and one diagonal is twice the length of the other, we can calculate the diagonals using trigonometry. Let ( d_1 ) be the longer diagonal and ( d_2 ) be the shorter diagonal. [ d_1 = 2 \times \text{side length of the rhombus} = 2 \times 4 = 8 ] [ d_2 = 2 \times \sin\left(\frac{3\pi}{8}\right) \times \text{side length of the rhombus} ] [ d_2 = 2 \times \sin\left(\frac{3\pi}{8}\right) \times 4 ] Calculate ( d_2 ) using trigonometric functions.

  2. Area of Each Triangular Face: Since the peak of the pyramid is directly above the center of the base, each triangular face of the pyramid is an isosceles triangle. The area of an isosceles triangle can be calculated using the formula: ( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} ). The base of each triangular face is the side length of the rhombus (4), and the height can be found using trigonometry. The height is one-half of the pyramid's height (since the peak is directly above the center of the base). [ \text{Height of each triangular face} = \frac{1}{2} \times \text{pyramid's height} = \frac{1}{2} \times 5 = 2.5 ]

  3. Surface Area: Once you have the area of the base and the area of each triangular face, you can find the total surface area by summing these areas.

[ \text{Total Surface Area} = \text{Area of Base} + 4 \times \text{Area of Each Triangular Face} ]

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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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