Question

1. a small box of chocolates is in the shape of a triangular prism with a base - units and a height of 10 units. the length of each side of the box is 12 units and the width of each side of a box is 9 feet. what is the surface area of the cardboard box of chocolates?

Explanation:

Step1: Find area of triangular bases

The base of the triangular - prism is an equilateral triangle with side length \(a = 9\) feet. The area of an equilateral triangle is \(A_{triangle}=\frac{\sqrt{3}}{4}a^{2}\). So \(A_{triangle}=\frac{\sqrt{3}}{4}\times9^{2}=\frac{81\sqrt{3}}{4}\) square feet. Since there are 2 triangular bases, the combined area of the bases is \(2\times\frac{81\sqrt{3}}{4}=\frac{81\sqrt{3}}{2}\) square feet.

Step2: Find area of rectangular faces

The rectangular faces have dimensions. The length of the prism \(l = 12\) feet and the width of each rectangular face is equal to the side - length of the triangular base \(a = 9\) feet. The area of one rectangular face is \(A_{rect}=l\times a=12\times9 = 108\) square feet. Since there are 3 rectangular faces, the combined area of the rectangular faces is \(3\times108 = 324\) square feet.

Step3: Calculate total surface area

The total surface area \(A\) of the triangular prism is the sum of the area of the bases and the area of the rectangular faces. \(A=\frac{81\sqrt{3}}{2}+324\).

$$A=\frac{81\times1.732}{2}+324$$

$$A=\frac{140.292}{2}+324$$

$$A = 70.146+324$$

$$A=394.146$$

square feet.

Answer:

\(394.146\) square feet