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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here's the solution to question 9: a) Calculate the total surface area of the prism. Step 1: Calculate the length of side BC using the Pythagorean theorem. In the right-angled triangle ABC ( ABC = 90^): AB^2 + BC^2 = AC^2 Given AB = 13 cm and AC = 20 cm. 13^2 + BC^2 = 20^2 169 + BC^2 = 400 BC^2 = 400 - 169 BC^2 = 231 BC = sqrt(231) ≈ 15.1987 cm Step 2: Calculate the area of the two triangular bases (ABC and DEF). The area of a right-angled triangle is (1)/(2) × base × height. Area_ ABC = (1)/(2) × AB × BC Area_ ABC = (1)/(2) × 13 cm × sqrt(231) cm Area_ ABC ≈ (1)/(2) × 13 × 15.1987 ≈ 98.7916 cm^2 The total area of the two bases is 2 × 98.7916 = 197.5832 cm^2. Step 3: Calculate the area of the three rectangular faces. The height of the prism is CF = 24 cm. Area of face ABED: Area_ABED = AB × AD = 13 cm × 24 cm = 312 cm^2 Area of face BCFE: Area_BCFE = BC × CF = sqrt(231) cm × 24 cm ≈ 15.1987 cm × 24 cm ≈ 364.7688 cm^2 Area of face ACFD: Area_ACFD = AC × CF = 20 cm × 24 cm = 480 cm^2 Step 4: Calculate the total surface area. Total Surface Area = (Area of 2 bases) + Area_ABED + Area_BCFE + Area_ACFD Total Surface Area = 197.5832 + 312 + 364.7688 + 480 Total Surface Area = 1354.352 cm^2 Rounding to three significant figures: Total Surface Area = 1350 cm^2 b) Calculate the angle that AF makes with the base BCPE. Step 1: Identify the relevant triangle and projection. The angle that line AF makes with the base BCPE (which is rectangle BCFE) is the angle between AF and its projection onto the base. Since AB is perpendicular to the base (as ABC = 90^ and it's a right prism), the projection of A onto the base is B. Therefore, the projection of AF onto the base is BF. The angle required is AFB. Step 2: Calculate the length of BF. BF is the diagonal of the rectangle BCFE. In the right-angled triangle BCF: BF^2 = BC^2 + CF^2 We know BC^2 = 231 and CF = 24 cm. BF^2 = 231 + 24^2 BF^2 = 231 + 576 BF^2 = 807 BF = sqrt(807) ≈ 28.4077 cm Step 3: Calculate AFB. Consider the right-angled triangle ABF, where ABF = 90^. We have AB = 13 cm (opposite to AFB) and BF = sqrt(807) cm (adjacent to AFB). Using the tangent function: ( AFB) = OppositeAdjacent = (AB)/(BF) ( AFB) = (13)/(sqrt(807)) ( AFB) ≈ (13)/(28.4077) ≈ 0.45761 AFB = ^-1(0.45761) AFB ≈ 24.59^ Rounding to one decimal place: AFB = 24.6^ Drop the next question.