Mathematics Benchmarks, Grades K-12

Leo's Painting

Rationale

This task allows students to integrate algebra and geometry as they generate a quadratic inequality from a proportional relationship.

Instructional Task

Leo is painting a picture on a canvas that measures 32 inches by 20 inches. He has divided the canvas into four unequally sized rectangles. In the lower left corner (see diagram), Leo wants to paint a person, while in the upper right corner he wants to paint another significant object (a globe, for example, as in the diagram). Because Leo knows that figures drawn in the golden ratio are aesthetically pleasing to the human eye, he would like the upper right corner to be a golden rectangle (that is, a rectangle whose length is about 1.6 times its width, found by the ratio $fraction: 1 + square root of 5 over 2$ ). Leo wants the area of the larger rectangle in the lower left to be at least half the total area of the canvas. Finally, he wants the upper right and lower left rectangles to share a corner, as seen in the diagram.

diagram

Find the range of possibilities for the dimensions of the smaller rectangle to the nearest hundredth.
If Leo uses the largest possible dimensions for the smaller rectangle,
a. what will the dimensions of the larger rectangle be?

b. will the larger rectangle also be "golden?"

c. is the original canvas a golden rectangle?

Discussion/Further Questions/Extensions

For more information on the golden rectangle, you may want to visit math websites such as mathworld.wolfram.com/GoldenRectangle.html.

In the sample solution, the value of w that fits the situation is given to be 5.85. Note that this is an approximate value, with the screen showing w = 5.8578644 (sample solution 1a). Since we are looking for values less than w, we have truncated this value to 5.85. This can be the source for an interesting discussion about when you might want to look at a value larger or smaller than an approximate value.

Sample Solutions

diagram

Find the range of possibilities for the dimensions of the smaller rectangle to the nearest hundredth.
To solve this problem, either algebraically or by using a graphing calculator, set it up to find an appropriate inequality:

l = 1.6w

Total area = 32 · 20 = 640 square inches

Area of the larger rectangle = (32 – 1.6w) (20 – w) = 1.6w² – 64w + 640

Because Leo wants the area of this larger rectangle to be at least half of the total canvas area, he sets up the following inequality:

1.6w² – 64w + 640 ≥ 320
1. Possible solution using the graph on a graphing calculator as a starting point:
  Graph the inequality 1.6w² – 64w + 640 ≥ 320 on a calculator, as shown below.
  
  For w values less than 5.85 or greater than 34.14, the y values on the parabola will be greater than 320. The value of w cannot be greater than 34.14, since this is larger than the original width (20 inches) of the canvas.
  
  Therefore, the width must be no larger than 5.85. Since l = 1.6w, or l = 1.6 (5.85), the length should be less than or equal to 9.36.
  
  w ≤ 5.85 in. and l ≤ 9.36 in.
2. Possible solution approaching the problem algebraically as a starting point:
  1.6w² – 64w + 640 ≥ 320
  
  1.6w² – 64w + 320 ≥ 0
  
  First solve the equation 1.6w² – 64w + 320 = 0 using the quadratic formula:
  
  $w = fraction: -(-64) plus or minus square root of (-64)squared - 4(1.6)(320) over 2(1.6)$
  
  w = 5.85 in. or 34.14 in.
  
  At this point you may want to refer to the graph or substitute values for w to see which solution region correctly meets the situation requirements. One possibility is that w ≤ 5.85 or w ≥ 34.14. Alternatively, it may be the case that 5.85 ≤ w ≤ 34.14. To determine which is the case, choose a value of w that falls within the indicated domain. Calculate the value of the left side of the original inequality to determine whether it yields a correct or an incorrect inequality.
  
  For example, let w = 15 or any number between 5.85 and 34.14.
  
  1.6w² – 64w + 640 ≥ 320 → 1.6(15)² – 64(15) + 640 ≥ 320 →
  
  The left side of this inequality is equal to 40, which is not greater than 320.
  
  As a second example, let w = 5 or any number less than 5.85.
  
  1.6w² – 64w + 640 ≥ 320 → 1.6(15)² – 64(15) + 640 ≥ 320 →
  
  The left side of this inequality is equal to 360, which is greater than 320. Therefore, the correct set of possible values for w is w ≤ 5.85 or w ≥ 34.14. Since 34.14 is larger than the original width (20 inches) of the canvas, w cannot be greater than or equal to 34.14. Therefore, the width must be less than or equal to 5.85.
  
  Since l = 1.6w, or l = 1.6 (5.85), the length is less than or equal to 9.36.
  
  w ≤ 5.85 in. and l ≤ 9.36 in.
If Leo uses the largest possible dimensions for the smaller rectangle,
1. what will the dimensions of the larger rectangle be?
  l = 32 – 9.36 = 22.64 and w = 20 – 5.85 = 14.15
  
  The dimensions of the larger rectangle are 22.64 in. by 14.15 in.
2. will the larger rectangle also be "golden?"
  The larger rectangle is "golden," since . Remember that in a golden rectangle, the length to width ratio is about 1.6.
3. is the original canvas a golden rectangle?
  The original canvas is a golden rectangle since .