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 questions 8, 9, and 10. 8. A classroom measures (x+2)m by (x-5)m. If the area of the classroom is 60m^2. Find its dimensions. Step 1: Set up the equation for the area of the classroom. (x+2)(x-5) = 60 Step 2: Expand the left side of the equation. x^2 - 5x + 2x - 10 = 60 x^2 - 3x - 10 = 60 Step 3: Rearrange the equation into a standard quadratic form ax^2 + bx + c = 0. x^2 - 3x - 10 - 60 = 0 x^2 - 3x - 70 = 0 Step 4: Factor the quadratic equation. We need two numbers that multiply to -70 and add to -3. These numbers are -10 and 7. (x-10)(x+7) = 0 Step 5: Solve for x. x-10 = 0 or x+7 = 0 x = 10 or x = -7 Step 6: Choose the valid value for x. Since dimensions cannot be negative, x=10 is the valid solution. Step 7: Calculate the dimensions using x=10. Length: x+2 = 10+2 = 12 m Width: x-5 = 10-5 = 5 m The dimensions of the classroom are 12 m by 5 m. 9. A bag contains 3 red balls and 4 blue balls. A ball is picked at random: Total number of balls = 3 (red) + 4 (blue) = 7 balls. a) Find the probability of picking red ball? Step 1: Identify the number of red balls and the total number of balls. Number of red balls = 3 Total number of balls = 7 Step 2: Calculate the probability. P(red) = Number of red ballsTotal number of balls = (3)/(7) The probability of picking a red ball is (3)/(7). b) Probability of picking blue ball? Step 1: Identify the number of blue balls and the total number of balls. Number of blue balls = 4 Total number of balls = 7 Step 2: Calculate the probability. P(blue) = Number of blue ballsTotal number of balls = (4)/(7) The probability of picking a blue ball is (4)/(7). c) Express the probability picking a red ball as a percentage? Step 1: Use the probability of picking a red ball from part a). P(red) = (3)/(7) Step 2: Convert the fraction to a percentage by multiplying by 100%. (3)/(7) × 100\% = (300)/(7)\% Step 3: Calculate the percentage. (300)/(7)\% ≈ 42.857\% Rounded to one decimal place, the probability is 42.9\%. 10. Solve using substitution: 2x - y = 6 (Equation 1) x - y = 1 (Equation 2) Step 1: From Equation 2, express x in terms of y. x - y = 1 x = y + 1 (Equation 3) Step 2: Substitute Equation 3 into Equation 1. 2(y+1) - y = 6 Step 3: Solve for y. 2y + 2 - y = 6 y + 2 = 6 y = 6 - 2 y = 4 Step 4: Substitute the value of y back into Equation 3 to find x. x = 4 + 1 x = 5 The solution is x=5, y=4. Last free one today — make it count tomorrow, or type /upgrade for unlimited.