A special box designed to hold an antique artifact is shaped like a triangular prism. The surface area of the box is 421.2 square inches. The height of the base triangle is 7.8 inches and each side of the base triangle is 9 inches long. What is the height of the box?

The area of the base is:

A = root ((s-a) * (s-b) * (s-c) * (s))

Where,

a, b, c: sides of the triangle

s = (a + b + c) / 2

We have then:

s = (9 + 9 + 9) / 2

s = 13.5

A = root ((13.5-9) * (13.5-9) * (13.5-9) * (13.5))

A = 35.07

Then, the surface area of the prism is:

S. A = 2 * A + 9h + 9h + 9h

Where,

h: height of the prism:

Substituting values:

421.2 = 2 * (35.07) + 9h + 9h + 9h

Clearing h:

27h = (421.2 - 2 * (35.07))

h = (421.2 - 2 * (35.07)) / (27)

h = 13

Answer:

the height of the box is:

h = 13 inches