Question

11 a small box of chocolates is in the shape of a triangular prism with a base of 15 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: Calculate base - area

The base is a triangle with sides \(a = 12\), \(b = 13\), \(c = 15\). First, find the semi - perimeter \(s=\frac{a + b + c}{2}=\frac{12+13 + 15}{2}=\frac{40}{2}=20\). Then, by Heron's formula, the area of the base \(B=\sqrt{s(s - a)(s - b)(s - c)}=\sqrt{20(20 - 12)(20 - 13)(20 - 15)}=\sqrt{20\times8\times7\times5}=\sqrt{5600}=20\sqrt{14}\).

Step2: Calculate lateral - area

The lateral faces are rectangles. There are three lateral faces with dimensions: one with dimensions \(10\times12\), one with \(10\times13\), and one with \(10\times15\). The lateral - area \(L=(10\times12)+(10\times13)+(10\times15)=10\times(12 + 13+15)=10\times40 = 400\).

Step3: Calculate surface - area

The surface - area \(S\) of a triangular prism is \(S = 2B+L\). Substitute \(B = 20\sqrt{14}\) and \(L = 400\) into the formula. \(S=2\times20\sqrt{14}+400=40\sqrt{14}+400\).

Answer:

\(40\sqrt{14}+400\) square units