We are given the following wooden box.
The sides, top, and bottom of the box are 1 centimeter thick. Let's look at the box when we include thickness.
Now, let's divide the box into two types of prisms:
- a bottom and a top small prisms with a pentagon in the base,
- five congruent prisms with a trapezoid in the base.
To find the volume of the given box, first, we will find a formula for the side of a pentagon with a known . Then, we will find the volumes of the two small pentagonal solids, bottom and top. After that, we will find the volume of a trapezoidal solid. At the end, we will calculate the volume of the box.
Formula for the Side of Pentagon
Now, we will find a formula for the side of a pentagon with a known apothem. Let a denotes the apothem.
To find AB in terms of a, let's divide the pentagon into five congruent isosceles triangles.
Notice the following.
- The angle ∠ AFB is a central angle, so m∠ AFB= 360^(∘)5=72^(∘).
- Since FG is an apothem, it is an altitude of isosceles Δ AFB. This tells us that FG bisects ∠ AFB. Therefore, m∠ AFG= 12m∠ AFB=36^(∘).
- By the , the sum of the angle measures of right Δ GAF is 180^(∘).
m∠ GAF+m∠ AFG+m∠ FGA=180^(∘)
m∠ GAF+ 36^(∘)+ 90^(∘)=180^(∘)
m∠ GAF=54^(∘)
cot(m∠ GAF)=AG/FG
cot 54^(∘)=AG/a
acot 54^(∘)=AG
AG=1.5cot 54^(∘)
AG≈ a( 0.7265)
AG≈ 0.7265a
Volume of Pentagonal Solid
Now, we will find the volume of one of two small pentagonal solids.
Since the sides, top, and bottom of the box are 1 centimeter thick, the height of the small prism is h= 1 cm. Now, let's analyze the base of this solid.
The apothem of the big pentagon is 4 cm and the thickness of wood is 1 cm. This tells us that the apothem of the small pentagon is a= 4- 1= 3 cm. Therefore, from Formula for the Side of Pentagon we can find a side of the small pentagon, s.
s=1.453 a
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Substitute 3 for a and evaluate
s=4.359
B=1/2aP
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Substitute values and evaluate
B=1/2( 3)( 21.795)
B=1/2(65.385)
B=1(65.385)/2
B=65.385/2
B=32.6925
V=Bh
▼
Substitute values and evaluate
V=32.6925
Volume of Trapezoidal Solid
Now, we will find the volume of one of five trapezoidal solids.
The height of this solid is h= 6 cm. Now, let's analyze the base of this solid.
The apothem of the big pentagon is 4 cm and the thickness of wood is 1 cm. This tells us that the apothem of the small pentagon is 4- 1=3 cm. Therefore, from Formula for the Side of Pentagon we can find AB and CD. Let's do it!
| Pentagon
|
Apothem
|
1.453*Apothem
|
Side
|
| Small
|
3
|
1.453(3)= 4.359
|
CD= 4.359
|
| Big
|
4
|
1.453(4)= 5.812
|
AB= 5.812
|
Now, to find the area of the base, let's use the formula for the , the height of the trapezoid is H= 1 cm.
B=1/2 H( AB+ CD)
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Substitute values and evaluate
B=1/2( 1)( 5.812+ 4.359)
B=1/2(1)(10.171)
B=1/2(10.171)
B=1(10.171)/2
B=10.171/2
B=5.0855
V=Bh
▼
Substitute values and evaluate
V=30.513
Volume of Box
The wooden box is made of 2 small pentagonal solids and 5 trapezoidal solids. We know that the volume of each pentagonal solid is V_1=32.6925 cubic centimeters, and the volume of each trapezoidal solid is V_2=30.513 cubic centimeters. Now, let's find the volume of the box.
V=2V_1+5V_2
▼
Substitute values and evaluate
V=2(32.6925)+5(30.513)
V=65.385+152.565
V=217.95
