Final answer:
The lateral area of a regular triangular right prism with a base edge of 10 cm and height 13 cm is 390 cm², and the total surface area is 476.6 cm² when rounded to the nearest tenth.
Step-by-step explanation:
To find the lateral area and surface areas of a regular triangular right prism, we use the formulas for the areas of triangles and rectangles. The base of the prism is an equilateral triangle with a side length of 10 cm. Since it's a right prism, the height of the triangular faces is the same as the length of the prism's side, which is 13 cm.
The lateral area of a prism is the perimeter of the base times the height of the prism (L.A. = perimeter x height). The perimeter of an equilateral triangle is 3 times the length of a side, so here it is 3 x 10 cm = 30 cm. Therefore, the lateral area is 30 cm x 13 cm = 390 cm².
For the surface area, we need to add the areas of the two triangular bases to the lateral area. The area of a triangle is 1/2 x base x height. For an equilateral triangle, the height can be found using the Pythagorean theorem, splitting the triangle into two 30-60-90 right triangles. This gives us a height of (sqrt(3)/2) x 10 cm = 8.66 cm (approx). So, the area of one triangular base is 1/2 x 10 cm x 8.66 cm = 43.3 cm². With two triangular bases, their combined area is 2 x 43.3 cm² = 86.6 cm². Adding this to the lateral area, the total surface area of the prism is 390 cm² + 86.6 cm² = 476.6 cm² (rounded to the nearest tenth).