A pyramid has a parallelogram shaped base and a peak directly above its center. Its base's sides have lengths of #2 # and #1 # and the pyramid's height is #2 #. 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 = 7.4788

#CH = 1 * sin ((pi)/6) = 0.5#
Area of parallelogram base #= a * b1 = 2*0.5 = color(red)(1)#

#EF = h_1 = sqrt(h^2 + (a/2)^2) = sqrt(2^2+ (2/2)^2)= 2.2361#
Area of # Delta AED = BEC = (1/2)*b*h_1 = (1/2)*1* 2.2361= #color(red)(1.1181)#

#EG = h_2 = sqrt(h^2+(b/2)^2 ) = sqrt(2^2+(1/2)^2 )= 2.1213#
Area of #Delta = CED = AEC = (1/2)*a*h_2 = (1/2)*2*2.1213 = color(red)( 2.1213)#

Lateral surface area = #2* DeltaAED + 2*Delta CED#
#=( 2 * 1.1181)+ (2* 2.1213) = color(red)(6.4788)#

Total surface area =Area of parallelogram base + Lateral surface area # = 1 + 6.4788 = 7.4788#

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Answer 2

The surface area of the pyramid can be calculated using the formula:

Surface Area = Base Area + (1/2) * Perimeter of Base * Slant Height

Given that the base is a parallelogram with side lengths of 2 and 1, and one of the base's corners has an angle of (5π)/6, the base area can be calculated as the product of the base's side lengths times the sine of the given angle.

Base Area = (2 * 1) * sin((5π)/6)

The perimeter of the base can be calculated by summing the lengths of the sides.

Perimeter of Base = 2 + 2 + 1 + 1 = 6

The slant height of the pyramid can be found using the Pythagorean theorem, considering that the slant height is the hypotenuse of a right triangle formed by the height of the pyramid and half of the diagonal of the base.

Slant Height = √(2^2 + (1/2)^2) = √(4 + 1/4) = √(17/4)

Now, plug the values into the formula for the surface area:

Surface Area = (2 * 1) * sin((5π)/6) + (1/2) * 6 * √(17/4)

Calculate the values and you'll get the 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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