A “wet floor” warning sign is in the shape of a triangular pyramid without a base. The edge length of the base equilateral triangle is 15 inches, the height of the base triangle is 13 inches, and the height of the triangular faces is 18 inches. How much fabric is needed to make the sign given the base does not need fabric?
To find the total surface area of the triangular pyramid, we need to calculate the area of the three triangular faces and the base.
The area of an equilateral triangle with side length 15 inches is:
$A_{base} = \frac{\sqrt{3}}{4} \times (side length)^2$
$A_{base} = \frac{\sqrt{3}}{4} \times 15^2$
$A_{base} = \frac{\sqrt{3}}{4} \times 225$
$A_{base} = \frac{225\sqrt{3}}{4}$
$A_{base} = \frac{675\sqrt{3}}{4}$
The area of each triangular face is:
$A_{face} = \frac{1}{2} \times base \times height$
$A_{face} = \frac{1}{2} \times 15 \times 18$
$A_{face} = \frac{1}{2} \times 15 \times 18$
$A_{face} = \frac{1}{2} \times 270$
$A_{face} = 135$
So, the total area of the triangular faces is:
$A_{total faces} = 3 \times A_{face}$
$A_{total faces} = 3 \times 135$
$A_{total faces} = 405$
Therefore, the total surface area of the sign is:
$A_{total} = A_{base} + A_{total faces}$
$A_{total} = \frac{675\sqrt{3}}{4} + 405$
$A_{total} = \frac{675\sqrt{3} + 1620}{4}$
$A_{total} = \frac{675\sqrt{3} + 1620}{4}$
Given that we need fabric on both sides of the sign, the amount of fabric required to make the sign is double the surface area:
$Fabric = 2 \times A_{total}$
$Fabric = 2 \times \frac{675\sqrt{3} + 1620}{4}$
$Fabric = \frac{1350\sqrt{3} + 3240}{4}$
$Fabric = \frac{1350\sqrt{3} + 3240}{4}$
Therefore, approximately 3375.75 square inches of fabric are needed to make the sign.