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.
ScanSolve AI Answer
Step 1: Define the dimensions of the rectangular courtyard. Let the length of the courtyard be L meters. Let the width of the courtyard be W meters. Given: L = (x+2) m and W = (x-4) m. Step 2: Use the given area to form an equation. The area of a rectangle is A = L × W. Given A = 40 m^2. 40 = (x+2)(x-4) Step 3: Expand and simplify the equation. 40 = x^2 - 4x + 2x - 8 40 = x^2 - 2x - 8 Rearrange into a standard quadratic form: x^2 - 2x - 8 - 40 = 0 x^2 - 2x - 48 = 0 Step 4: Solve the quadratic equation for x. We need two numbers that multiply to -48 and add to -2. These numbers are -8 and 6. (x-8)(x+6) = 0 Step 5: Find the possible values for x. Set each factor to zero: x-8 = 0 x = 8 x+6 = 0 x = -6 Step 6: Choose the valid value for x. Since dimensions (length and width) cannot be negative, x must be positive. If x=-6, the width (x-4) would be -10, which is not possible. Therefore, x=8. Step 7: Calculate the length and width of the courtyard. Length L = (x+2) = (8+2) = 10 m. Width W = (x-4) = (8-4) = 4 m. The length of the courtyard is 10 m and the width is 4 m. Step 1: Define variables based on the initial situation. Let x be the original number of women in the group. The total amount to be contributed is sh 1200. The original contribution per woman would be (1200)/(x) shillings. Step 2: Define variables based on the changed situation. Five women were unable to contribute, so the new number of women is x-5. The remaining women contributed sh 20 more than the original amount per woman. So, the new contribution per woman is ((1200)/(x) + 20) shillings. Step 3: Formulate the equation. The total amount contributed by the remaining women is still sh 1200. (New number of women) × (New contribution per woman) = Total amount (x-5)((1200)/(x) + 20) = 1200 Step 4: Expand the equation. x((1200)/(x)) + x(20) - 5((1200)/(x)) - 5(20) = 1200 1200 + 20x - (6000)/(x) - 100 = 1200 Step 5: Simplify and rearrange the equation. 1100 + 20x - (6000)/(x) = 1200 Subtract 1200 from both sides: 20x - (6000)/(x) - 100 = 0 Step 6: Multiply the entire equation by x to eliminate the fraction. 20x^2 - 6000 - 100x = 0 Step 7: Rearrange into the standard quadratic form ax^2 + bx + c = 0. 20x^2 - 100x - 6000 = 0 Step 8: Divide by the common factor (20) to simplify the quadratic equation. (20x^2)/(20) - (100x)/(20) - (6000)/(20) = 0 x^2 - 5x - 300 = 0 The quadratic equation representing the information is x^2 - 5x - 300 = 0. 3 done, 2 left today. You're making progress.