The problem asks to find the surface area of a triangular prism. The dimensions are given in centimeters (cm). The triangular base has a base of 4 cm and a height of 4 cm. The two other sides of the triangle are 4.5 cm each. The length of the prism is 6 cm.

1. Problem Description

The problem asks to find the surface area of a triangular prism. The dimensions are given in centimeters (cm). The triangular base has a base of 4 cm and a height of 4 cm. The two other sides of the triangle are 4.5 cm each. The length of the prism is 6 cm.

2. Solution Steps

The surface area of the triangular prism can be found by summing the areas of its faces. The prism has two triangular faces and three rectangular faces.

Area of a triangle:

$A_{triangle} = \frac{1}{2} \times base \times height$

Area of a rectangle:

$A_{rectangle} = length \times width$

Area of one triangular face:

$A_{triangle} = \frac{1}{2} \times 4 \ cm \times 4 \ cm = 8 \ cm^2$

Since there are two identical triangular faces, the total area of the two triangles is $2 \times 8 \ cm^2 = 16 \ cm^2$.

The rectangular faces have dimensions 6 cm x 4 cm, 6 cm x 4.5 cm, and 6 cm x 4.5 cm.

Area of the first rectangular face:

$A_{rectangle1} = 6 \ cm \times 4 \ cm = 24 \ cm^2$

Area of the second rectangular face:

$A_{rectangle2} = 6 \ cm \times 4.5 \ cm = 27 \ cm^2$

Area of the third rectangular face:

$A_{rectangle3} = 6 \ cm \times 4.5 \ cm = 27 \ cm^2$

Total area of the three rectangular faces is $24 \ cm^2 + 27 \ cm^2 + 27 \ cm^2 = 78 \ cm^2$.

The total surface area of the triangular prism is the sum of the areas of the two triangles and the three rectangles.

$A_{total} = 16 \ cm^2 + 78 \ cm^2 = 94 \ cm^2$

3. Final Answer

94 cm^2