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.
what is the volume of metal in the hex nut, to the nearest tenth? use 3.14 for $pi$.
$circ$ $21.1\\ \text{cm}^3$
$circ$ $23.6\\ \text{cm}^3$
$circ$ $27.6\\ \text{cm}^3$
$circ$ $31.6\\ \text{cm}^3$
area of a regular hexagon $=\frac{1}{2}$(apothem)(perimeter)

Explanation:

Step1: Find hexagon side length

A regular hexagon can be divided into 6 equilateral triangles. The apothem $a = 2$ cm is the height of one equilateral triangle. For an equilateral triangle with side length $s$, height $a = \frac{\sqrt{3}}{2}s$, so:
$$s = \frac{2a}{\sqrt{3}} = \frac{2\times2}{\sqrt{3}} = \frac{4}{\sqrt{3}} \approx 2.31 \text{ cm}$$
(Matches the given 2.3 cm for approximation)

Step2: Calculate hexagon perimeter

Perimeter $P = 6s = 6\times2.3 = 13.8$ cm

Step3: Find hexagon base area

Use given formula: $A_{hex} = \frac{1}{2} \times \text{apothem} \times \text{perimeter}$
$$A_{hex} = \frac{1}{2} \times 2 \times 13.8 = 13.8 \text{ cm}^2$$

Step4: Calculate volume of hexagonal prism

Prism height $h = 2$ cm, so $V_{prism} = A_{hex} \times h$
$$V_{prism} = 13.8 \times 2 = 27.6 \text{ cm}^3$$

Step5: Find cylinder hole radius

Diameter = 1.6 cm, so radius $r = \frac{1.6}{2} = 0.8$ cm

Step6: Calculate volume of cylinder hole

$V_{cylinder} = \pi r^2 h$, use $\pi=3.14$
$$V_{cylinder} = 3.14 \times (0.8)^2 \times 2 = 3.14 \times 0.64 \times 2 = 4.0192 \text{ cm}^3$$

Step7: Compute volume of metal

Subtract cylinder volume from prism volume: $V_{metal} = V_{prism} - V_{cylinder}$
$$V_{metal} = 27.6 - 4.0192 = 23.5808 \approx 23.6 \text{ cm}^3$$

Answer:

23.6 cm³