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alexandr1967 [171]
1 year ago
9

A steel hex nut has two regular hexagonal bases and a cylindrical hole with a diameter of 1.6 centimeters through the middle. th

e apothem of the hexagon is 2 centimeters. a cylinder is cut out of the middle of a hexagonal prism. the hexagon has an apothem with a length of 2 centimeters and base side lengths of 2.3 centimeters. the prism has a height of 2 centimeters. the cylinder has a diameter of 1.6 centimeters. the equation for the area of a regular hexagon = one-half (apothem) (perimeter). what is the volume of metal in the hex nut, to the nearest tenth? use 3.14 for π.
Mathematics
1 answer:
Margaret [11]1 year ago
5 0

Subtracting the volume of the cylinder from the volume of the prism, the volume of metal in the hex nut to the nearest tenth exists $$23.6 cm^3

<h3>How to estimate the volume of metal in the hex nut?</h3>

Diameter of the cylinder be d = 1.6 cm

Apothem of the hexagon be a = 2 cm

Thickness of the steel hex nut be t = 2 cm

Volume of the prism be V_p

Volume of the cylinder be V_c

Volume of metal in the hex nut,

$$V = V_p - V_c

To estimate the volume of a prism,

$$V_p = A_b h

Ab = n L a / 2

Number of the sides, n = 6

The side of the hexagon be L

Height of the prism, h = t = 2 cm

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

A = 360°/6 = 60°

$tan (\frac{A}{2} )=(\frac{L/2}{a})

simplifying the value of L, we get

$tan (\frac{60}{2} )=(\frac{L/2}{2})

$tan 30}=(\frac{L/2}{2})

$tan (\frac{\sqrt{3}}{3} )=(\frac{L/2}{2})

Solving for L/2:

$\frac{2  \sqrt{3}}{3}  =\frac{L}{2}

Solving the value of L, we get

$2\frac{2  \sqrt{3}}{3}  =L

$\frac{4  \sqrt{3}}{3}  =L

$L=4 \sqrt{3}/3 cm

Ab = n L a / 2

Substitute the values in the above equation, we get

$A_b=\frac{6 (4 \sqrt{3}/3)(2)}{2}

$$A_b=24 \sqrt{3}/3 $$cm^2

$A_b=8 \sqrt{3} cm^2

V_p = A_b h

substitute the values in the above equation, we get

$V_p=(8 \sqrt{3})(2)

$V_p=16 \sqrt{3} cm^3

$$V_p=16 (1.732) cm^3

$$V_p=27.712 cm^3

To estimate the volume of cylinder,

$V_c = (πd^2/4) h

Here, π = 3.14 and d = 1.6 cm

Height of the cylinder, h = t = 2 cm

substitute the values in the above equation, we get

$V_c=[3.14 (1.6) / 4] (2)

$V_c=[3.14 (2.56) / 4] (2)

$V_c=(2.0096) (2)

$$V_c=4.019 cm^3

Substitute the values in the equation, we get

$$V=V_p-V_c

$$V=27.712 - 4.019

$$V=23.693 cm^3

$$V=23.6 cm^3

Therefore, the volume of metal in the hex nut, to the nearest tenth exists $$23.6 cm^3.

To learn more about volume of cylinder and prism refer to:

brainly.com/question/12669337

#SPJ4

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Step-by-step explanation:

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