A pyramid has a parallelogram shaped base and a peak directly above its center. Its base's sides have lengths of #6 # and #5 # and the pyramid's height is #6 #. If one of the base's corners has an angle of #(5pi)/6#, what is the pyramid's surface area?

Answer 1

T S A = 87.541

#CH = 5 * sin (pi/6) = 2.5#
Area of parallelogram base #= a * b1 = 6*2.5 = color(red)(15 )#

#EF = h_1 = sqrt(h^2 + (a/2)^2) = sqrt(6^2+ (6/2)^2)= 6.7082#
Area of # Delta AED = BEC = (1/2)*b*h_1 = (1/2)*5* 6.7082= #color(red)(16.7705)#

#EG = h_2 = sqrt(h^2+(b/2)^2 ) = sqrt(6^2+(5/2)^2 )= 6.5#
Area of #Delta = CED = AEC = (1/2)*a*h_2 = (1/2)*6*6.5 = color(red)( 19.5)#

Lateral surface area = #2* DeltaAED + 2*Delta CED#
#=( 2 * 16.7705)+ (2* 19.5) = color(red)(72.541)#

Total surface area =Area of parallelogram base + Lateral surface area # = 15 + 72.541 = 87.541#

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Answer 2

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

Given that the base is a parallelogram, its area can be calculated using the formula: ( \text{Area of parallelogram} = \text{base} \times \text{height} ).

Given the base sides as 6 and 5, and the height as 6, the area of the base is (6 \times 6 = 36) square units.

To find the area of each triangular face, you first need to calculate the lengths of the slant heights. The slant height can be found using the Pythagorean theorem.

The slant height, (l), can be calculated as (l = \sqrt{h^2 + (\frac{b}{2})^2}), where (h) is the height of the pyramid (6) and (b) is the length of the base side (5 or 6).

For each triangular face, the area can be calculated using the formula for the area of a triangle: ( \text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height} ).

Substitute the calculated slant height into the formula for the area of each triangular face. There are four triangular faces.

After calculating the areas of all four triangular faces, add them to the area of the base to find the total surface area of the pyramid.

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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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