Question

Navibulgar is a bulgarian shipping company. the department that handles fragile shipping rolls prints and other documents into cylinders and then pack the cylinders into special containers that are triangular solids so that the cylinders would not roll around in the trailers. the surface area of a triangular prism container that would be needed to enclose a rolled document with a diameter of 10cm and a length of 85 cm? Round your answer to the nearest square centimeter. Assume the triangle face is an equilateral triangle

Answer 1

Answer:

4677 Square Centimeters

Step-by-step explanation:

Diameter of the Triangle =10cm, therefore:

Radius of the triangle =5cm

The Cylindrical containers are enclosed in Triangular prisms where the triangular face is assumed to be an Equilateral Triangle.  

The cross section is given in Figure 1.  

In the Equilateral Triangle ABC, points P, Q and R are the points of tangency.

We apply the following theorems:

Theorem 1: The angle between a tangent and a radius is 90 degrees

Theorem 2: Tangents to a circle from the same point are equal.

In Right Triangle BPO, presented in Figure 2, From Figure 1, |BO| divides the angle at B into two equal parts of 30 degrees each since it is an equilateral triangle.

Therefore, Angle PBO=30 degrees.

Sin 30 =|PO|/|BO|

0.5=5/BO|

BO|=5/0.5=10cm

Similarly

Tan 30=5/|PB|

|PB|=5/Tan 30=5√3 cm

By the same process,

|AR|=5√3 cm

Theorem: The altitude of an equilateral triangle divides the base into two equal parts.

Therefore: |AR|=|RC|=5√3 cm

Height of the Triangle |BR|=|BO|+|OR|=10+5=15cm

Base of the Triangle, |AC| =5√3+5√3=10√3 cm

Area of the Triangle =0.5*15*10√3 =75√3 Square cm

The Prism has two equilateral triangular faces and three rectangular faces of the same dimension. See Figure 4.

Therefore:

Surface Area of the Triangular Prism=Area of Two Triangular Faces+Area of Three Rectangular Faces

=2(75√3)+3(10√3*85)

=4676.5 Square Centimeters

=4677 Square Centimeters