Asteel hex nut has two regular hexagonal bases and a cylindrical hole with a diameter of 1.6 centimeters through the middle. the apothem of the hexagon is 2 centimeters. what is the volume of metal in the hex nut, to the nearest tenth? use 3.14 for π. 21.1 cm3 23.6 cm3 27.6 cm3 31.6 cm3

Subtracting the volume of the cylinder from the volume of the prism, the volume of metal in the hex nut to the nearest tenth is 23.6 cm^3 (second option)

Step-by-step explanation:

Diameter of the cylinder: d=1.6 cm

Apothem of the hexagon: a=2 cm

Assuming the thickness of the steel hex nut: t=2 cm

Volume of metal in the hex nut: V=?

V=Vp-Vc

Volume of the prism: Vp

Volume of the cylinder: Vc

Prism:

Vp=Ab h

Ab=n L a / 2

Number of the sides: n=6

Side of the hexagon: L

Height of the prism: h=t=2 cm

Central angle in the hexagon: A=360°/n

A=360°/6

A=60°

tan (A/2)=(L/2) / a

tan (60°/2)=(L/2) / (2 cm)

tan 30° = (L/2) / (2 cm)

sqrt(3)/3=(L/2) / (2 cm)

Solving for L/2:

(2 cm) sqrt(3)/3 = L/2

2 sqrt(3)/3 cm = L/2

Solving for L:

2 (2 sqrt(3)/3 cm)=L

4 sqrt(3)/3 cm = L

L=4 sqrt(3)/3 cm

Ab=n L a / 2

Ab=6 (4 sqrt(3)/3 cm)(2 cm) / 2

Ab=24 sqrt(3)/3 cm^2

Ab=8 sqrt(3) cm^2

Vp=Ab h

Vp=(8 sqrt(3) cm^2)(2 cm)

Vp=16 sqrt(3) cm^3

Vp=16 (1.732) cm^3

(1) Vp=27.712 cm^3

Cylinder:

Vc=(π d^2/4) h

π=3.14

d=1.6 cm

Height of the cylinder: h=t=2 cm

Vc=[3.14 (1.6 cm)^2 / 4] (2 cm)

Vc=[3.14 (2.56 cm^2) / 4] (2 cm)

Vc=(2.0096 cm^2) (2 cm)

Vc=4.019 cm^3

V=Vp-Vc

V=27.712 cm^3 - 4.019 cm^3

V=23.693 cm^3

V=23.6 cm^3

898.1666667

step-by-step explanation:

5389/6 = 898.1666667

step-by-step explanation:

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