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
a) Step 1: Factorize the quadratic expression for the area. The area of the rectangular garden is given by A = x^2 - 4x - 21. To factorize this quadratic, we need to find two numbers that multiply to -21 and add up to -4. These numbers are 3 and -7. So, the expression can be factored as: A = (x + 3)(x - 7) Step 2: Determine the dimensions of the garden. For a rectangle, Area = Length × Width. Therefore, the dimensions of the garden are the factors we found. Length = (x + 3) units Width = (x - 7) units Since dimensions must be positive, x-7 > 0, which means x > 7. The dimensions of the garden are (x+3) and (x-7). b) Step 1: Calculate the perimeter of the garden. The perimeter P of a rectangle with length L and width W is given by P = 2(L + W). Using the dimensions from part a): P = 2((x + 3) + (x - 7)) P = 2(x + 3 + x - 7) P = 2(2x - 4) P = (4x - 8) meters Step 2: Calculate the cost of fencing. The cost of fencing wire is Ksh 1.50 per meter. Total cost = Perimeter × Cost per meter Total cost = (4x - 8) × 1.50 Total cost = 1.5(4x - 8) Total cost = (6x - 12) Ksh The cost of fencing the garden is (6x - 12) Ksh$. 3 done, 2 left today. You're making progress.