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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You're on a roll — here are the solutions for questions 4 and 5: 4. Step 1: Write down the formula for the area of a trapezium. The area A of a trapezium with parallel sides a and b and perpendicular height h is given by: A = (1)/(2)(a+b)h Step 2: Identify the given values. The parallel sides are PQ = 8 cm and SR = 6 cm. So, a=8 and b=6. The area A = 28 cm^2. We need to find the perpendicular distance h. Step 3: Substitute the values into the formula. 28 = (1)/(2)(8+6)h 28 = (1)/(2)(14)h Step 4: Simplify the equation. 28 = 7h Step 5: Solve for h. Divide both sides by 7: h = (28)/(7) h = 4 cm The perpendicular distance between PQ and SR is: 4 cm 5. To determine the least mass of tea leaves in the bag, we need to find the Least Common Multiple (LCM) of the possible packet sizes: 40 g, 250 g, and 350 g. This is because the total mass must be divisible by each of these amounts. Step 1: Find the prime factorization of each number. 40 = 2 × 2 × 2 × 5 = 2^3 × 5^1 250 = 2 × 5 × 5 × 5 = 2^1 × 5^3 350 = 2 × 5 × 5 × 7 = 2^1 × 5^2 × 7^1 Step 2: Calculate the LCM by taking the highest power of each prime factor present in any of the numbers. The prime factors are 2, 5, and 7. Highest power of 2: 2^3 (from 40) Highest power of 5: 5^3 (from 250) Highest power of 7: 7^1 (from 350) Step 3: Multiply these highest powers together to find the LCM. LCM(40, 250, 350) = 2^3 × 5^3 × 7^1 = 8 × 125 × 7 = 1000 × 7 = 7000 The least mass of the tea leaves in the bag is: 7000 grams Got more? Send 'em!