Final answer:
To find the prism's surface area, calculate the area of the triangular base, determine the height using the volume, then sum the area of the two bases and the three rectangles formed by the sides of the prism.
Step-by-step explanation:
We are given a prism with a volume of 540dm³ and a triangular base with sides measuring 5 dm, 12 dm, and 13 dm. To find the surface area of the prism, we first need to calculate the area of the triangular base using Heron's formula, which requires the semi-perimeter (s) of the triangle:
s = (5dm + 12dm + 13dm) / 2 = 15dm
Next, calculate the area (A) of the triangle:
A = √[s(s - 5dm)(s - 12dm)(s - 13dm)]
Plug in the values and calculate to find the area of the triangle, which will be the base area (B) of the prism.
Once the base area (B) is known, we use the formula for the volume of the prism (V = B x h) to find the height (h) of the prism by rearranging it to h = V / B. After calculating h, we can find the surface area of the prism, which is the sum of the areas of the two triangular bases and three rectangles that correspond to the sides of the prism. The surface area (SA) can be calculated using:
SA = 2B + (perimeter of the base x height of the prism)
Calculate the perimeter of the base, multiply it by the height, add twice the base area, and you will have the total surface area of the prism.