A pyramid has a parallelogram shaped base and a peak directly above its center. Its base's sides have lengths of #2 # and #3 # and the pyramid's height is #7 #. If one of the base's corners has an angle of #(5pi)/6#, what is the pyramid's surface area?

Question

A pyramid has a parallelogram shaped base and a peak directly above its center.  Its base's sides have lengths of #2 # and #3  # and the pyramid's height is #7 #.  If one of the base's corners has an angle of #(5pi)/6#, what is the pyramid's surface area?

Maverick Smith · Accepted Answer

T S A = 38.5312

#CH = 2 * sin ((5pi)/6) = 1#
Area of parallelogram base #= a* b1 = 3*1 = color(red)(3)#

#EF = h_1 = sqrt(h^2 + (a/2)^2) = sqrt(7^2+ (3/2)^2)= 7.1589#
Area of # Delta AED = BEC = (1/2)*b*h_1 = (1/2)*2* 7.1589= #color(red)(7.1589)#

#EG = h_2 = sqrt(h^2+(b/2)^2 ) = sqrt(7^2+(2/2)^2 )= 7.0711#
Area of #Delta = CED = AEC = (1/2)*a*h_2 = (1/2)*3*7.0711 = color(red)( 10.6067)#

Lateral surface area = #2* DeltaAED + 2*Delta CED#
#=( 2 * 7.1589)+ (2* 10.6067) = color(red)(35.5312)#

Total surface area =Area of parallelogram base + Lateral surface area # = 3 + 35.5312 = 38.5312#

Aurora Ayala · Answer

undefined To find the surface area of the pyramid, we need to calculate the area of the base and the lateral area.

1. Area of the base:
Since the base is a parallelogram, its area is given by the formula: $ \text{Area of parallelogram} = \text{base} \times \text{height} $.
Given the base's sides are 2 and 3, the area of the base is $ 2 \times 3 = 6 $ square units.

2. Lateral area:
The lateral area of a pyramid is given by the formula: $ \text{Lateral area} = \frac{1}{2} \times \text{perimeter of base} \times \text{slant height} $.
To find the perimeter of the base, we add the lengths of the four sides: $ 2 + 3 + 2 + 3 = 10 $.
Now, we need to find the slant height. We can use trigonometry with the given angle.
The slant height ($ l $) is given by $ l = \frac{\text{height}}{\sin(\text{angle})} $.
Given the height is 7 and the angle is $ \frac{5\pi}{6} $, $ l = \frac{7}{\sin\left(\frac{5\pi}{6}\right)} $.
Using trigonometric identities, we know $ \sin\left(\frac{5\pi}{6}\right) = \sin\left(\pi - \frac{\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} $.
So, $ l = \frac{7}{\frac{1}{2}} = 14 $.
Now, plug these values into the formula to find the lateral area: $ \text{Lateral area} = \frac{1}{2} \times 10 \times 14 = 70 $ square units.

3. Total surface area:
The total surface area is the sum of the base area and the lateral area: $ \text{Total surface area} = \text{Base area} + \text{Lateral area} = 6 + 70 = 76 $ square units.

So, the surface area of the pyramid is $ 76 $ square units.