Answer:
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
