Question

A company sells tents in two styles, shown below. The sides and floor of each tent are made of nylon.

Which tent requires less nylon to manufacture? Use 3.14 for pi and round to the hundredths place. Show all work to verify your conclusion.

Answer 1

Prism shaped tent will require more nylon.

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 +
`2* (√(3))/(4)(\text{side})^(2)`

= 3(8²) +
`(√(3))/(2)* 64`

= 192 + 55.43

= 247.43 ft²

Now surface area of the cylindrical tent =
`\pi rh+2((1)/(2)\pi r^(2))+(Side)^(2)`

=
`\pi (4)* 8+\pi (4^(2))+8^(2)`

= 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.

Answer 2

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:

`B =(\pi r^2)/(2)` 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.

`B =(\pi (4)^2)/(2) = 8 \pi`

Area of square =
`(side)^2`

then;

Area of square =
`(8)^2 = 64 ft` square ft

Lateral surface area of cylinder =
`2\pi rh` where r is the radius and h is the height

Here, h = 8 ft

Lateral surface area of half cylinder =
`\pi rh = \pi (4)(8) = 32 \pi` 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;

`S = 2 \cdot (8 \pi) +32 \pi + 64 = 64 +48 \pi = 64+ 48(3.14) = 214.72` 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 =
`(side)^2`

then;

Area of square =
`(8)^2 = 64 ft` square ft

Area of a triangle is given by:

`A = (1)/(2) b \cdot h`

Each side of a triangle is same.

This is an equilateral triangle, and its height is given by:

`h = (√(3) )/(2) \text{side}`

then;

`h = (√(3) )/(2) \cdot 8 = 4 √(3)`

Area of triangle =
`(1)/(2) (8) \cdot 4 √(3) =16 √(3)` square ft.

`\text{Total surface area of Pup tent} = 3 \cdot \text{Area of a square} + 2 \cdot \text{Area of a triangle}`

Substitute the given values we have;

Total surface area of Pup tent(S')

=
`3 \cdot 64 + 2 \cdot (16√(3)) = 192+32√(3)=247.4` square ft.

⇒S' > S

therefore, the tunnel tent require less nylon to manufacture.