Respuesta :
Answer:
As per the statement:
A company sells tents in two styles, shown below.
The sides and floor of each tent made of nylon.
For tunnel tent:
In a composite figure of tunnel tent :
There are two semi circle, half cylindrical shape and its base is made of Square.
Area of semi-circle is given by:
[tex]B =\frac{\pi r^2}{2}[/tex] where r is the radius the circle.
Here, Diameter(d) = 8 feet
We know that: Diameter(d) = 2 r
then;
8 = 2r
⇒ r= 4 ft.
[tex]B =\frac{\pi (4)^2}{2} = 8 \pi[/tex]
Area of square = [tex](side)^2[/tex]
then;
Area of square = [tex](8)^2 = 64 ft[/tex] square ft
Lateral surface area of cylinder = [tex]2\pi rh[/tex] where r is the radius and h is the height
Here, h = 8 ft
Lateral surface area of half cylinder = [tex]\pi rh = \pi (4)(8) = 32 \pi[/tex] square ft
Total Surface area of tunnel tent(S)=2(Area of semi-circle)+(Lateral surface area of half cylindrical)+(Area of square)
Substitute the given values we have;
[tex]S = 2 \cdot (8 \pi) +32 \pi + 64 = 64 +48 \pi = 64+ 48(3.14) = 214.72[/tex] square ft.
For Pup tent:
Each side of tent = 8 ft
In a figure of Pup tent:
There are 3 square shape and two triangle.
Area of square = [tex](side)^2[/tex]
then;
Area of square = [tex](8)^2 = 64 ft[/tex] square ft
Area of a triangle is given by:
[tex]A = \frac{1}{2} b \cdot h[/tex]
Each side of a triangle is same.
This is an equilateral triangle, and its height is given by:
[tex]h = \frac{\sqrt{3} }{2} \text{side}[/tex]
then;
[tex]h = \frac{\sqrt{3} }{2} \cdot 8 = 4 \sqrt{3}[/tex]
Area of triangle =[tex]\frac{1}{2} (8) \cdot 4 \sqrt{3} =16 \sqrt{3}[/tex] square ft.
[tex]\text{Total surface area of Pup tent} = 3 \cdot \text{Area of a square} + 2 \cdot \text{Area of a triangle}[/tex]
Substitute the given values we have;
Total surface area of Pup tent(S')
= [tex]3 \cdot 64 + 2 \cdot (16\sqrt{3}) = 192+32\sqrt{3}=247.4[/tex] square ft.
⇒S' > S
therefore, the tunnel tent require less nylon to manufacture.
Answer:
Prism shaped tent will require more nylon.
Step-by-step explanation:
Amount of nylon required will be decided by both the shapes shown in the figure.
First shape of tent is in the shape of a prism having all sides 8 ft in measure.
Since total surface area of the tent including surface and sides
= Area of all square sides + area of triangular sides
= 3(Side)² + [tex]2\times \frac{\sqrt{3}}{4}(\text{side})^{2}[/tex]
= 3(8²) + [tex]\frac{\sqrt{3}}{2}\times 64[/tex]
= 192 + 55.43
= 247.43 ft²
Now surface area of the cylindrical tent = [tex]\pi rh+2(\frac{1}{2}\pi r^{2})+(Side)^{2}[/tex]
= [tex]\pi (4)\times 8+\pi (4^{2})+8^{2}[/tex]
= 32π + 16π + 64
= 48π + 64
= 214.86 ft²
Since surface area of the prism shaped ten is higher than cylindrical tent, therefore, more nylon will be required to prepare a prism shaped tent.
