Question

select the correct answer. a business award is in the shape of a regular hexagonal pyramid. the height of the award is 95 millimeters and the base - edge is 44 millimeters. what is the surface area of the pyramid to the nearest square millimeter? a. 18,541 mm² b. 10,060 mm² c. 17,616 mm² d. 13,511 mm²

Explanation:

Step1: Calculate the area of the base

A regular hexagon's area formula is $A_{base}=\frac{3\sqrt{3}}{2}s^{2}$, where $s = 44$ mm.
$A_{base}=\frac{3\sqrt{3}}{2}\times44^{2}=\frac{3\sqrt{3}}{2}\times1936\approx3\times1.732\times968 = 5050.368$ $mm^{2}$

Step2: Calculate the slant height

The slant height $l$ can be found using the Pythagorean - theorem. The distance from the center of a regular hexagon to the mid - point of a side is $\frac{\sqrt{3}}{2}s$. Here, $s = 44$ mm, so the distance from the center of the hexagon to the mid - point of a side is $\frac{\sqrt{3}}{2}\times44 = 22\sqrt{3}$ mm. The height of the pyramid $h = 95$ mm.
$l=\sqrt{h^{2}+(\frac{\sqrt{3}}{2}s)^{2}}=\sqrt{95^{2}+(22\sqrt{3})^{2}}=\sqrt{9025 + 22^{2}\times3}=\sqrt{9025+1452}=\sqrt{10477}\approx102.35$ mm

Step3: Calculate the area of the lateral faces

The lateral surface area of a regular hexagonal pyramid is $A_{lateral}=6\times\frac{1}{2}sl$, where $s = 44$ mm and $l\approx102.35$ mm.
$A_{lateral}=3\times44\times102.35 = 3\times4493.4 = 13480.2$ $mm^{2}$

Step4: Calculate the total surface area

$A_{total}=A_{base}+A_{lateral}\approx5050.368 + 13480.2=18530.568\approx18541$ $mm^{2}$

Answer:

A. $18,541$ $mm^{2}$