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2.7 – Mathematical Modeling MODELING VOLUME An open box with a square base is to be cut from a square piece of cardboard with side equal to 18 inches but cutting out a square from each corner and turning up the sides. Draw a Diagram: Write the volume, V, of the box as a function of the length x of the square cut from the box. Write in standard form: For what value of x is the volume the largest? What is the largest possible volume? MODELING AREA A farmer has 3000 feet of fencing to enclose a rectangular field. One side lies along a river, so only three sides need fencing. Draw a diagram: Express the area A of the field enclosed by the fence as function of l, the length of the side parallel to the river. For what value of l is the area the largest? What is the largest area? A rectangle is inscribed in a semicircle of radius 2. Let P = (x, y) be the point in Quadrant I that is a vertex of the rectangle and is on the circle. Express the area A of the rectangle as a function of x. Express the perimeter P as a function of x. For what value of x is A the largest? How about P?