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 variables and set up equations. Let the length of the rectangular enclosure be L meters and the width be W meters. The total length of wire mesh is the perimeter of the rectangle. 2L + 2W = 100 The area of the rectangular enclosure is given by: A = L × W Step 2: Express the area in terms of a single variable. From the perimeter equation, we can simplify: L + W = 50 Express L in terms of W: L = 50 - W Substitute this into the area equation: A = (50 - W)W A = 50W - W^2 Step 3: Find the width that maximizes the area. To find the maximum area, we differentiate the area function A with respect to W and set the derivative to zero. (dA)/(dW) = (d)/(dW)(50W - W^2) (dA)/(dW) = 50 - 2W Set (dA)/(dW) = 0: 50 - 2W = 0 2W = 50 W = 25 m To confirm this is a maximum, we can check the second derivative: (d^2A)/(dW^2) = -2 Since the second derivative is negative, the value W=25 m corresponds to a maximum area. Step 4: Calculate the corresponding length and the maximum area. Substitute W = 25 m back into the equation for L: L = 50 - W = 50 - 25 = 25 m Now, calculate the maximum area: A_max = L × W = 25 m × 25 m A_max = 625 m^2 The greatest area the farmer can enclose is 625 m^2. Last free one today — make it count tomorrow, or type /upgrade for unlimited.