Question

The Problem: Depress the equation r6r +100 1. Decomposing a cube: Consider a cube with side length (a) Suppose we break the side of the cube at an arbitrary point ryb. This cut triggers the decomposition of the cube into the 8 pieces you have with your manipulative. You will have a cube with side length y and a cube with side length b. Identify the other 6 solids in terms of their dimensions using y and b so that we have the lengths b (b) Regroup the 8 pieces into four solids by grouping pieces of the same dimensions. Find the volumes of these four solids. Notice that the volume of the entire solid is = (y+b= sum of the volumes of the solids. Find the sum of the volumes of the four solids. What do you notice about this algebraic identity? 2. Decomposing a rectangular prism: Consider a second solid with square base z, height 6, and total volume 6z2. b This solid can be decomposed (like in problem 1) by cutting the edge of length z to form four pieces (see the figure above). That is, the side of the solid is being split into the lengths I= y+b. If we regroup the pieces into three solids by grouping pieces of the same dimension, abed then we obtain solids with volume 6y2, 12by, and 6b2. Make sure you understand why this is true. Notice that the volume of the entire solid is 6a2 - 6(y+b)2= 3. Depressing a cube: Recall that our goal is to solve the equation r36x2 + 100. (a) Describe (in words) the relationship between the equation r362+ 100 and the volumes from problems 1 and 2. 6y2 + 12by + 6b2 +100 (b) Explain why we can say y +3by2 +3b2yb 4. At this point, the cut in the solids from Problems 1 and 2 has been arbitrary. We would like to determine the location of the cut so that the solid with volume 3by2 in Part 1b and the solid with volume 6y2 from problem 2 are equivalent. What value of b will make these volumes equivalent? (Note: This value will be the location of the cut to decompose the cube and the rectangular prism. 5. Why do we want 3by2 =6y2 in problem 4? [Hint: Consider the goal of this activity 6. Simplify the equality from problem 3b using the value of b from problem 4 (write your equality using only positive coefficients). What does the simplified equality represent geometrically? Be as specific possible. as 7. What is special about the simplified equality from problem 6? going to (briefly) describe how to depress a cubic equation (of the form absent in class. What would you tell this 8. Imagine you were x3=Cr D) geometrically to a student who was student?

Answer 1

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