Quadratic Function Application Problems

Quadratic and Cubic Function Max-Min Application Problems

1. Find the dimensions and area of the largest rectangle that can be constructed in the region bounded by the x-axis and the graph of the parabola y = 9 - x² with one side of the rectangle on the x-axis and the vertices of the opposite side lying on the graph of the parabola. Click here to see an animation showing the graph of the parabola in blue, a sequence of the inscribed rectangles in red, and the area function in red (shifted four units to the right) with an animated point moving along the graph of the area function that corresponds to the area of each rectangle as it is drawn. Quicktime animation.

The approximate dimensions of the largest rectangle would be 3.4641 by 6.	The red portion of the graph above shows the area function.
	The graph above shows the largest possible rectangle inscribed between the parabola and the x-axis.

2. A total of 8 meters of fencing are going to be used to fence in a rectangular cage for pets and divide it into three smaller cages as shown in the animation. Determine the overall dimensions that will yield the maximum total enclosed area. In the animation the total area function is graphed in red. The blue point moving along the area function corresponds to the changing size of the rectangular cage. The perimeter of the cage is shown in blue and the added dividers in green. Let x stand for the length of one of the sides (with 2 sides needed of length x) and y stand for the length of the other side (with 4 sections of fencing needed of length y). Quicktime Animation

	The red portion of the graph above shows the area function.
	The picture above shows the base of the cage divided into three cages. Click the picture to see an animation.

3. A rectangular piece of material measuring 4 ft by 3 ft is to be formed into an open topped box by cutting equal sized squares out of each corner and folding up the sides. Determine the size of the squares to be cut out if the resulting box is to have the maximum possible volume. Click here to see an animation with scales or on the figure at the right below to see an animation without scales. Click here for a 3-D animation without scales and click here for a 3-D animation with scales. In the animation we see the changing shape of the material after various sized squares are cut out of the corners along with the volume function in red shifted to the right. The 3-D animations show the changing shape of the box after the sides have been folded up. The animated point moving along the graph of the volume function corresponds to the changing box construction. Quicktime animation 3D Quicktime animation DPGraph animation Here is a more psychedelic DPGraph animation. Here are two DPGraph pictures of the box. These can be looked at for entertainment for 15 minutes or so if you have no life. Picture of the box one Picture of the box two (this is picture one with shading)

	The red portion of the graph above shows the Volume function.
	The picture above shows the rectangular piece of material with squares cut out of each corner. Click the picture to see an animation.

4. Find the rectangle of maximum area that can be constructed with one vertex at the origin, one vertex on the positive x-axis, one vertex on the positive y-axis, and one vertex on the graph of the line with equation y = (6-x)/2. Geometer's Sketchpad Video

	Click on the picture to see an animation showing an animated rectangle along with a corresponding point moving along the graph of the area function. Quicktime Version
	The red portion of the graph above shows the area function.

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Lane Vosbury, Math Chair, Seminole Community College email: vosburyl@scc-fl.edu