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 are the solutions to the problems. Problem 1: Prism a) Calculate the volume of the prism. Step 1: Calculate the area of the triangular base. The base of the triangle is 12 cm and the height is 5 cm. Area of base = (1)/(2) × base × height = (1)/(2) × 12 cm × 5 cm = 30 cm^2 Step 2: Calculate the volume of the prism. The length of the prism is 25 cm. Volume = Area of base × length = 30 cm^2 × 25 cm = 750 cm^3 b) Calculate the total surface area of the prism. Step 1: Calculate the area of the two triangular bases. Area of two bases = 2 × ((1)/(2) × 12 cm × 5 cm) = 60 cm^2 Step 2: Calculate the length of the hypotenuse of the triangular base. Using the Pythagorean theorem: Hypotenuse = sqrt(12^2 + 5^2) = sqrt(144 + 25) = sqrt(169) = 13 cm Step 3: Calculate the area of the three rectangular faces. The length of the prism is 25 cm. Area of rectangle 1 = 12 cm × 25 cm = 300 cm^2 Area of rectangle 2 = 5 cm × 25 cm = 125 cm^2 Area of rectangle 3 = 13 cm × 25 cm = 325 cm^2 Step 4: Calculate the total surface area. Total surface area = 60 cm^2 + 300 cm^2 + 125 cm^2 + 325 cm^2 = 810 cm^2 Problem 2: Marks scored by 20 pupils The marks are: 10, 11, 12, 15, 8, 11, 16, 10, 12, 10, 11, 12, 8, 10, 16, 10, 8, 10, 8, 12. a) Complete the frequency table below. Step 1: Count the frequency for each mark. • Mark 8: 4 times • Mark 10: 6 times • Mark 11: 3 times • Mark 12: 4 times • Mark 15: 1 time • Mark 16: 2 times Step 2: Calculate f × x for each mark and find the totals. |c|c|c| Marks, x & Frequency, f & f × x \\ 8 & 4 & 32 \\ 10 & 6 & 60 \\ 11 & 3 & 33 \\ 12 & 4 & 48 \\ 15 & 1 & 15 \\ 16 & 2 & 32 \\ Total & f = 20 & (f × x) = 220 \\ b) Find the mode mark. Step 1: Identify the mark with the highest frequency. From the frequency table, the mark 10 has the highest frequency of 6. The mode mark = 10 c) Calculate the mean mark. Step 1: Use the formula for the mean of grouped data. Mean = ( (f × x))/( f) Step 2: Substitute the total values from the frequency table. Mean = (220)/(20) = 11 Send me the next one 📸