Optimization Exploration: The Inscribed Rectangle. Learning Objectives: Materials:

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1 Optimization Exploration: The Inscribed Rectangle Lesson Information Written by Jonathan Schweig and Shira Sand Subject: Pre-Calculus Calculus Algebra Topic: Functions Overview: Students will explore some classic optimization problems using The Geometer s Sketchpad. This lesson has students begin by exploring a pattern, then has them extrapolate their findings into a generalization about inscribed shapes. Extension questions are included. Technology: Geometry Software Level: Difficult Activity Structure: Self-Guided Problem Solving Duration of Activity: Whole Class Period Multiple Classes Learning Objectives: Students will learn how to find the length and width of the largest rectangle that can be inscribed under a parabola Students will learn how to make a general claim about the maximum dimensions of any inscribed rectangle. Students will use their findings to solve similar problems. Materials: The Geometer s Sketchpad Software GSP Worksheet entitled Inscribed Rectangle Paper Pencil Worksheet

2 Problem: 2 A rectangle is bounded by the x-axis and the parabola y = x + 9. What length and width should the rectangle have so that its area is maximized? Exploration: 1. Drag point A to change the dimensions of the rectangle. Observe the changes in the base and height of the rectangle as point A moves. a) What happens to the base of the rectangle as the x-coordinate of point A changes? b) What happens to the height of the rectangle as the x-coordinate of point A changes? 2. Where do you think point A should be in order to create the desired rectangle? 3. Click on Show Table. This table shows the area of a rectangle for a particular abscissa (x-coordinate) of point A (now called x A ). Change the dimensions of the rectangle by dragging point A. How do the numbers in the table change? 4. Create a table for five different values of x A, including the one you proposed in question two. Copy your table below.

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4 5. Is the rectangle you proposed in question two the rectangle with the largest area on your table? What does this tell you about the desired rectangle? 6. Plot a rough sketch of area vs. abscissa of A on the coordinate axes below. 7. What type of mathematical relationship does this resemble? 8. Click on Show graph and then click on Change dimensions. Watch the trace of the plotted point as the dimensions of the rectangle change. What type of mathematical relationship does this resemble now? Does this match your answer from question 7?

5 9. What point on the graph represents the maximum area of the rectangle? 10. Drag point A to the location you guessed in question two. Is the plotted point at the maximum value of your graph? 11. Click on Move to maximum area. How far was your hypothesized point from the desired point (in terms of its x-coordinate)? 12. Describe the shape of the rectangle with the maximum area.

6 Generalization: 1. Consider a parabola given by the equation bounded by the x-axis and this parabola. 2 y = x + a. A rectangle is a. Describe the base of the rectangle in terms of the x-coordinate of its lower right-hand corner. b. Describe the height of the rectangle in terms of the x-coordinate of its lower right-hand corner. 2. Write a function of x representing the area of the rectangle. 3. What type of function is this? What do you know about the behavior of this type of function? 4. What is the domain of your function? 5. Were your predictions in questions 7 and 8 of the exploration correct? If not, explain what caused you to come to an incorrect conclusion. 6. Describe a method for finding the maximum value for this type of function.

7 Extensions: 2 1. A rectangle is bounded by the x-axis and the semicircle y = 25 x. What base and height should the rectangle have so that its area is maximized? 2. A rectangle is inscribed in a circle of radius 10. What base and height should the rectangle have so that its area is maximized? 3. A rectangle is inscribe in a right triangle having sides of lengths 6, 8, and 10 inches. Find the dimensions of the rectangle of greatest area. 4. Find the volume of the largest right circular cylinder that can be inscribed in a sphere of radius 10.