Question

a prism with a height of 14 millimeters has two regular hexagons for bases. the sides of the hexagon measure 7.5 millimeters, and the apothem of the hexagon measures 6.495 millimeters. what is the surface - area of the prism? 812.3 mm² 1214.8 mm² 355.3 mm² 728.4 mm²

Explanation:

Step1: Calculate base - area

The area formula for a regular polygon is $A = \frac{1}{2}aP$, where $a$ is the apothem and $P$ is the perimeter. For a regular hexagon with side - length $s = 7.5$ mm, the perimeter $P=6s = 6\times7.5=45$ mm and apothem $a = 6.495$ mm. The area of one base $A_{base}=\frac{1}{2}\times6.495\times45 = 146.1375$ mm². The area of two bases $A_{2bases}=2\times146.1375 = 292.275$ mm².

Step2: Calculate lateral - area

The lateral - area of a prism $A_{lateral}=Ph$, where $P$ is the perimeter of the base and $h$ is the height of the prism. We know $P = 45$ mm and $h = 14$ mm, so $A_{lateral}=45\times14=630$ mm².

Step3: Calculate surface - area

The surface - area of a prism $A_{surface}=A_{2bases}+A_{lateral}$. Substitute the values of $A_{2bases}$ and $A_{lateral}$: $A_{surface}=292.275 + 630=922.275\approx922.3$ mm².

Answer:

$922.3$ mm²