Respuesta :
Answer: The lower supports are Congruent Rectangles and the area of the two supports is 24 square meters.
The upper arch can be decomposed as one semicircle with radius 6 meters minus a semicircle with radius 3 meters.
The area of the archway is (13.5 π + 24) square meters.
Step-by-step explanation:
By the given diagram,
The area of the archway = The area of upper support + Area of lower support.
Since, the lower support are two congruent rectangles having dimension 3 m × 4 m
Hence, The area of lower support = 3 × 4 + 3 × 4 = 12 + 12 = 24 square m.
Now, the upper support is the common the bounded area by the two concentric semi circles having radius 3 m and 6 m,
Hence, the area of the upper support = Area of semi circle having radius 6 m - Area of semi circle having radius 3 m
[tex]=\frac{\pi(6)^2}{2}-\frac{\pi(3)^2}{2} = \frac{36\pi-9\pi}{2}=\frac{27\pi}{2}=13.5\pi\text{ square m}[/tex]
Thus, the area of the archway = (13.5 π + 24) square meters
The completed information is as follows;
- The lower support are same sized rectangles and the area of the two support is 24 square meters.
- The upper arch can be decomposed as one semicircle with radius 6 meters minus a semicircle with radius 3 meters.
- The area of the archway is [tex]\underline{(13.5 \cdot \pi + 24)}[/tex] square meters
Methods used for the calculations
- A description for the lower support = Same sized rectangles
- The area of the two supports = 2 × 3 m × 4 m = 24 m²
The radius of the large semicircle on the upper arch, R = Half the distance between the two support + The width of a pillar
Therefore;
R = 6 m ÷ 2 + 3 m = 6 m
- The radius of the large semicircle on the upper arch, R = 6 meters
The radius of the smaller semicircle, r = 3 meters
The area of the upper arc, A = Area of the larger semicircle - Area of the smaller semicircle
Area of a semi circle = π × (Radius)
Therefore;
[tex]A = \dfrac{1}{2} \times \pi \times R^2 - \dfrac{1}{2} \times \pi \times r^2 [/tex]
Which gives;
[tex]A = \dfrac{1}{2} \times \pi \times 6^2 - \dfrac{1}{2} \times \pi \times 3^2 = \dfrac{1}{2} \cdot \pi \cdot \left(36-9 \left) = \dfrac{27}{2} \cdot \pi = 13.5 \cdot \pi[/tex]
- The area of the upper arch, A = 13.5·π square meters
The area of the archway = Area of the upper arch + Area of the two pillars
Which gives;
- Area of the archway = [tex]\underline{(13.5 \cdot \pi + 24) }[/tex] square meters
Learn more about the areas of composite figures here:
https://brainly.com/question/16955380
