Final answer:
To find the dimensions that will give the rectangle of greatest area, solve the equation 2L + 2W = 40 for one variable, substitute it into the area formula, and find the maximum value of the area function.
Step-by-step explanation:
To find the dimensions that will give the rectangle of greatest area, we need to use the formula for the area of a rectangle, which is length multiplied by width. Let's assume the length of the rectangle is L and the width is W. The perimeter of the rectangle is given as 40, so we have the equation 2L + 2W = 40.
To find the dimensions that give the greatest area, we can solve this equation for one variable and substitute it into the area formula. Let's solve the equation for L: 2L = 40 - 2W, and divide both sides by 2: L = 20 - W. Now we substitute this expression for L into the area formula: A = (20 - W) * W = 20W - W^2.
To find the dimensions that give the greatest area, we need to find the maximum value of the area function A. We can do this by finding the vertex of the parabola defined by the area function. The vertex of a parabola is at the x-coordinate -b/2a, where a is the coefficient of the x^2 term, and b is the coefficient of the x term. In our case, a = -1 and b = 20, so the x-coordinate of the vertex is -20/(-2) = 10.
Thus, the width that gives the rectangle of greatest area is W = 10, and the corresponding length is L = 20 - W = 20 - 10 = 10.