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
Here's the solution to the rectangle problem: i) Step 1: Write the formula for the perimeter of a rectangle. The perimeter P of a rectangle with length L and width W is given by P = 2(L+W). Given L = (x+3) cm and W = y cm. Given P = 24 cm. Step 2: Substitute the given values into the perimeter formula. 2((x+3) + y) = 24 Step 3: Simplify the equation to show y = 9 - x. Divide both sides by 2: (x+3) + y = 12 x + 3 + y = 12 Subtract x and 3 from both sides: y = 12 - x - 3 y = 9 - x This shows the required relationship. ii) Step 1: Write the formula for the area of a rectangle. The area A of a rectangle with length L and width W is given by A = L × W. Given L = (x+3) cm and W = y cm. Given A = 27 cm^2. Step 2: Substitute the given values into the area formula. (x+3)y = 27 Step 3: Substitute y = 9 - x (from part i) into the area equation. (x+3)(9-x) = 27 Step 4: Expand and simplify the equation to show x^2 - 6x = 0. 9x - x^2 + 27 - 3x = 27 -x^2 + 6x + 27 = 27 Subtract 27 from both sides: -x^2 + 6x = 0 Multiply by -1: x^2 - 6x = 0 This shows the required relationship. iii) Step 1: Solve the quadratic equation from part ii for x. x^2 - 6x = 0 Factor out x: x(x - 6) = 0 This gives two possible values for x: x = 0 or x - 6 = 0 x = 6 Step 2: Use the equation y = 9 - x from part i to find the corresponding values of y. If x = 0: y = 9 - 0 = 9 If x = 6: y = 9 - 6 = 3 Step 3: Determine the dimensions for each possible value of x. Case 1: If x = 0 Length L = (x+3) = (0+3) = 3 cm. Width W = y = 9 cm. In this case, the "length" (3 cm) is shorter than the "width" (9 cm), which contradicts the usual convention that length is greater than or equal to width. Case 2: If x = 6 Length L = (x+3) = (6+3) = 9 cm. Width W = y = 3 cm. In this case, the "length" (9 cm) is greater than the "width" (3 cm), which is consistent with the usual convention. Step 4: Select the valid solution and state the width. Based on the convention that the length of a rectangle is typically greater than or equal to its width, Case 2 is the valid solution. Therefore, the width of the rectangle is 3 cm. The width of the rectangle is 3 cm. That's 2 down. 3 left today — send the next one.