This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Step 1: Identify the dimensions of the triangular prism. The prism has a right-angled triangular base. The base of the triangle is b = 5 cm. The height of the triangle is h = 10 cm. The length of the prism (depth) is L = 25 cm. Step 2: Calculate the area of the two triangular bases. The area of one triangular base is A_triangle = (1)/(2) × base × height. A_triangle = (1)/(2) × 5 cm × 10 cm = 25 cm^2 Since there are two triangular bases (front and back), their combined area is: A_two triangles = 2 × 25 cm^2 = 50 cm^2 Step 3: Calculate the length of the hypotenuse of the triangular base. Let the hypotenuse be c. Using the Pythagorean theorem: c^2 = b^2 + h^2 c^2 = (5 cm)^2 + (10 cm)^2 c^2 = 25 cm^2 + 100 cm^2 c^2 = 125 cm^2 c = sqrt(125) cm = sqrt(25 × 5) cm = 5sqrt(5) cm Step 4: Calculate the area of the three rectangular faces. • The bottom rectangular face has dimensions 5 cm × 25 cm. A_bottom = 5 cm × 25 cm = 125 cm^2 • The vertical side rectangular face has dimensions 10 cm × 25 cm. A_vertical side = 10 cm × 25 cm = 250 cm^2 • The slanted top rectangular face has dimensions 5sqrt(5) cm × 25 cm. A_slanted top = 5sqrt(5) cm × 25 cm = 125sqrt(5) cm^2 Step 5: Calculate the total surface area. The total surface area (TSA) is the sum of the areas of the two triangles and the three rectangular faces. TSA = A_two triangles + A_bottom + A_vertical side + A_slanted top TSA = 50 cm^2 + 125 cm^2 + 250 cm^2 + 125sqrt(5) cm^2 TSA = (50 + 125 + 250 + 125sqrt(5)) cm^2 TSA = (425 + 125sqrt(5)) cm^2 To get a numerical value, we approximate sqrt(5) ≈ 2.236: TSA ≈ (425 + 125 × 2.236) cm^2 TSA ≈ (425 + 279.5) cm^2 TSA ≈ 704.5 cm^2 The total surface area is 704.5 cm^2. That's 2 down. 3 left today — send the next one.