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 # pi/3 #. What is the pyramid's surface area?

Answer 1

T S A = 56.9376

AB = BC = CD = DA = a = 4
Height OE = h = 5
OF = a/2 = 4/2 = 2
# EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(5^2+2^2) = color(red)(5.3852)#

Area of #DCE = (1/2)*a*EF = (1/2)*4*5.3852 = color(red)(10.7704)#
Lateral surface area #= 4*Delta DCE = 4*10.7704 = color(blue)(43.0816)#

#/_C = (pi)/3#
Area of base ABCD #= a* a * sin /_C = 4^2 sin (pi/3) = 13.856#

T S A #= Lateral surface area + Base area#
T S A # =43.0816 + 13.856 = color(purple)(56.9376)#

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Answer 2

To find the surface area of the pyramid, we 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: Area = base * height. In this case, since the side length of the rhombus is given as 4, and the height can be calculated using trigonometry, as h = a * sin(θ), where a is the side length (4) and θ is the angle (π/3 radians). So, height of the rhombus = 4 * sin(π/3) = 4 * √3 / 2 = 2√3. Area of the rhombus = base * height = 4 * 2√3 = 8√3.

  2. Area of each triangular face: Each triangular face of the pyramid is an isosceles triangle with two equal sides (the slant height of the pyramid) and a base equal to one side of the rhombus (4 units). The slant height can be found using the Pythagorean theorem with the height (5 units) and half the side length of the rhombus (2 units). So, slant height (l) = √(5^2 + 2^2) = √29. Now, we can calculate the area of one triangular face using the formula: Area = 0.5 * base * height, where the base is 4 units and the height is the slant height we found. Area of one triangular face = 0.5 * 4 * √29 = 2√29.

  3. Total surface area: Since there are four triangular faces, the total surface area is the sum of the areas of the base and the four triangular faces. Total surface area = Area of base + 4 * Area of one triangular face Total surface area = 8√3 + 4 * 2√29 = 8√3 + 8√29 ≈ 45.44 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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