A pyramid has a parallelogram shaped base and a peak directly above its center. Its base's sides have lengths of #4 # and #1 # and the pyramid's height is #6 #. If one of the base's corners has an angle of #pi/4 #, what is the pyramid's surface area?

Answer 1

T S A = 33.2358

#CH = 1 * sin (pi/4) = 0.707#
Area of parallelogram base #= a * b1 = 4*0.707 = color(red)(2.828 )#

#EF = h_1 = sqrt(h^2 + (a/2)^2) = sqrt(6^2+ (4/2)^2)= 6.3246#
Area of # Delta AED = BEC = (1/2)*b*h_1 = (1/2)*1* 6.3246= #color(red)(3.1623)#

#EG = h_2 = sqrt(h^2+(b/2)^2 ) = sqrt(6^2+(1/2)^2 )= 6.0208#
Area of #Delta = CED = AEC = (1/2)*a*h_2 = (1/2)*4*6.0248 = color(red)( 12.0496)#

Lateral surface area = #2* DeltaAED + 2*Delta CED#
#=( 2 * 3.1623)+ (2* 12.0496) = color(red)(30.4078)#

Total surface area =Area of parallelogram base + Lateral surface area # = 2.828 + 30.4078 = 33.2358#

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Answer 2

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

[ \text{Surface Area} = \text{Base Area} + \text{Area of Four Triangular Faces} ]

The base area of the pyramid is the area of the parallelogram, which is given by:

[ \text{Base Area} = \text{Base Length} \times \text{Base Width} ]

In this case, the base length is 4 and the base width is 1.

Next, we need to find the area of each triangular face. Since the peak is directly above the center of the base, each triangular face is an isosceles triangle with two sides of length 6 (the height of the pyramid) and one side of length equal to the diagonal of the base parallelogram. The diagonal of a parallelogram can be found using the Pythagorean theorem.

The diagonal (d) of a parallelogram with sides of lengths (a) and (b) and angle (\theta) between them is given by:

[ d = \sqrt{a^2 + b^2 + 2ab\cos(\theta)} ]

In this case, (a = 4), (b = 1), and (\theta = \frac{\pi}{4}).

Once you find the diagonal, you can use the formula for the area of an isosceles triangle:

[ \text{Area of Triangle} = \frac{1}{2} \times \text{Base} \times \text{Height} ]

In this case, the base of the triangle is the diagonal of the base parallelogram and the height is the height of the pyramid.

Calculate the area of one triangular face, and then multiply by 4 to get the total area of all four triangular faces.

Add the base area and the total area of the triangular faces to get 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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