Concept explainers
Here we celebrate the power of algebra as a powerful way of finding unknown quantities by naming them, of expressing infinitely many relationships and connections clearly and succinctly, and of uncovering pattern and structure.
Box o’ knots (H). There is a box of knotted bunches of string in your math instructor’s classroom. The number of trefoil knots in the box is x and the number of figure-eight knots is y. You take all the knots out of the box and display them so each looks like the trefoil or figure-eight knot illustrated in Mindscape 20. If there are a total of 21 knots and 74 crossings, how many knots are there of each type?
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The Heart of Mathematics: An Invitation to Effective Thinking
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- Enclosing a field You have 16 miles of fence that you will use to enclose a rectangle field. a.Draw a picture to show that you can arrange the 16 miles fence into a rectangle of width 3 miles and length 5 miles. What is the area of this rectangle? b.Draw a picture to show that you can arrange the 16 miles fence into a rectangle of width 2 miles and length 6 miles. What is the area of this rectangle? c.The first two parts of this exercise are designed to show you that you can get different areas for the rectangle of the sane perimeter, 16 miles. In general, if you arrange the 16 miles of fence into a rectangle of width w miles, then it will enclose an area of A=w8-w square miles. iMake a graph of the area enclosed as a function of w, and explain what the graph is showing. iiWhat width w should you use to enclose the most area? iiiWhat is the length of the maximum-area rectangle that you made, and what kind of figure do you have?arrow_forwardA Precocious Child and Her Blocks A child has 64 blocks that are 1-inch cubes. She wants to arrange the blocks into a solid rectangle h blocks long and w blocks wide. There is a relationship between h and w that is determined by the restriction that all 64 blocks must go into the rectangle. A rectangle h blocks long and w blocks wide uses a total of h X w blocks. Thus, hw = 64. Applying some elementary algebra, we get the relationship that we need: w=64h(2.3) Use a formula to express the perimeter P in terms of h and w. Using Equation 2.3, find a formula that expresses the perimeter P in terms of the height only. How should the child arrange the blocks if she wants the perimeter to be the smallest possible? Do parts b and c again, this time assuming that the child has 60 blocks rather than 64 blocks. In this situation, the relationship between h and w is w = 60/h. Note: Be careful when you do part c. The child will not cut the blocks into piecesarrow_forward
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