Question

A steel 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. 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.

Answer 1

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 $23.6 cm^3$

How to estimate the volume of metal in the hex nut?

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,

$V_c$ = (π$d^2$/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

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