Answer true or false. A fence is to be used to enclose a rectangular plot of land. If there are 1360 feet of fencing, it can be shown that a 340 ft by 340 ft square is the rectangle that can be enclosed with the greatest area. (A square is considered a rectangle.)

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This is a classic optimization problem in which we want to maximize the area of a rectangular plot enclosed by a given amount of fencing. The formula for the perimeter of a rectangle is P = 2l + 2w, where l is the length and w is the width. We are given that there are 1360 feet of fencing, so we can set up the equation 1360 = 2l + 2w and solve for w in terms of l: w = 680 - l.

To maximize the area of the rectangle, we need to find the dimensions that give us the largest product of length and width. Substituting w = 680 - l into the formula for area A = lw, we get A = l(680 - l) = 680l - l^2. This is a quadratic function with a negative leading coefficient, so it has a maximum value at the vertex of the parabola. The x-coordinate of the vertex is given by x = -b/2a, where a = -1, b = 680, and c = 0 (since there is no constant term). Solving for x, we get x = 680/2 = 340. Therefore, the rectangle with dimensions 340 ft by 340 ft has the largest area that can be enclosed with 1360 feet of fencing.

Answer True or False