Jawaban:

A

Penjelasan dengan langkah-langkah:

The total area of the solid can be found by multiplying the number of pentagonal faces by the area of each pentagon. Since the solid is made using only pentagonal faces, the number of faces is equal to the number of pentagons used.

Let's assume that the solid has n pentagonal faces. To find n, we need to use Euler's formula, which states that for any convex polyhedron (a solid with flat faces), the number of vertices minus the number of edges plus the number of faces is equal to 2. Mathematically,

V - E + F = 2

where V is the number of vertices, E is the number of edges, and F is the number of faces.

Since the solid has only pentagonal faces, each face has 5 vertices, and each vertex is shared by 3 faces (as in a regular dodecahedron). Therefore, we can write:

V = (5n)/3

E = V/2 + 5n/2 = (3n + 5n)/2 = 4n

F = n

Substituting these values into Euler's formula, we get:

(5n/3) - 4n + n = 2

Simplifying the equation, we get:

n = 6

Therefore, the solid has 6 pentagonal faces. The total area of the solid is:

6 × 30√3 = 180√3 cm²

So the answer is (a) 270√3.