The Heart of Mathematics: An Invitation to Effective Thinking
4th Edition
ISBN: 9781118156599
Author: Edward B. Burger, Michael Starbird
Publisher: Wiley, John & Sons, Incorporated
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Textbook Question
Chapter 4.4, Problem 29MS
T-total (H). Suppose you start with one small tile. You create a super tile by surrounding the small tile with t additional small tiles identical to the original. Then you create a super-super tile by surrounding your super tile with t addit ional super tiles. Write an expression for the total number of small tiles you have in your super-super tile. Suppose this total is 81. Find the value of t.
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The college hiking club is having a fundraiser to buy new equipment for fall and winter outings. The club is selling Chinese fortune cookies at a price of $2 per cookie. Each cookie contains a piece of paper with a different number written on it. A random drawing will determine which number is the winner of a dinner for two at a local Chinese restaurant. The dinner is valued at $32. Since fortune cookies are donated to the club, we can ignore the cost of the cookies. The club sold 718 cookies before the drawing. Lisa bought 13 cookies. Lisa's expected earnings can be found by multiplying the value of the dinner by the probability that she will win. What are Lisa's expected earnings? Round your answer to the nearest cent.
What was the age distribution of nurses in Great Britain at the time of Florence Nightingale? Thanks to Florence Nightingale and the British census of 1851, we have the following information (based on data from the classic text Notes on Nursing, by Florence Nightingale). Note: In 1851 there were 25,466 nurses in Great Britain. Furthermore, Nightingale made a strict distinction between nurses and domestic servants. Use a histogram and graph the probability distribution. Using the graph of the probability distribution determine the probability that a British nurse selected at random in 1851 would be 40 years of age or older. Round your answer to nearest thousandth.
Age range (yr)
20–29
30–39
40–49
50–59
60–69
70–79
80+
Midpoint (x)
24.5
34.5
44.5
54.5
64.5
74.5
84.5
Percent of nurses
5.7%
9.7%
19.5%
29.2%
25.0%
9.1%
1.8%
What was the age distribution of nurses in Great Britain at the time of Florence Nightingale? Thanks to Florence Nightingale and the British census of 1851, we have the following information (based on data from the classic text Notes on Nursing, by Florence Nightingale). Note: In 1851 there were 25,466 nurses in Great Britain. Furthermore, Nightingale made a strict distinction between nurses and domestic servants. Use a histogram and graph the probability distribution. Using the graph of the probability distribution determine the probability that a British nurse selected at random in 1851 would be 40 years of age or older. Round your answer to nearest thousandth.
Age range (yr)
20–29
30–39
40–49
50–59
60–69
70–79
80+
Midpoint (x)
24.5
34.5
44.5
54.5
64.5
74.5
84.5
Percent of nurses
5.7%
9.7%
19.5%
29.2%
25.0%
9.1%
1.8%
Chapter 4 Solutions
The Heart of Mathematics: An Invitation to Effective Thinking
Ch. 4.1 - The main event. State the Pythagorean Theorem.Ch. 4.1 - Two out of three. If a right triangle has legs of...Ch. 4.1 - Hypotenuse hype. If a right triangle has legs of...Ch. 4.1 - Assesing area. Suppose you know the base of a...Ch. 4.1 - Squares all around. How does the figure below...Ch. 4.1 - Operating on the triangle. Using a straightedge,...Ch. 4.1 - Excite your friends about right triangles....Ch. 4.1 - Easy as 1,2,3? Can there be a right triangle with...Ch. 4.1 - Sky high (S). On a sunny, warm day, a student...Ch. 4.1 - Sand masting (H). The sailboat named Sand Bug has...
Ch. 4.1 - Getting a pole on a bus. For his 13th birthday,...Ch. 4.1 - The Scarecrow (ExH). In the 1939 movie The Wizard...Ch. 4.1 - Rooting through a spiral. Start with a right...Ch. 4.1 - Is it right? (H) Suppose someone tells you that...Ch. 4.1 - Tfrain trouble (H). Train tracks are made of...Ch. 4.1 - Does everyone have what it takes to be a triangle?...Ch. 4.1 - Getting squared away. In our proof of the...Ch. 4.1 - The practical side of Pythagoras. Suppose you are...Ch. 4.1 - Pythagorean pizzas (H). You have a choice at the...Ch. 4.1 - Natural right (S). Suppose r and s are any two...Ch. 4.1 - Well-rounded shapes. Suppose we have two circles...Ch. 4.1 - A Pythagorean Theorem for triangles other than...Ch. 4.1 - With a group of folks. In a small group, discuss...Ch. 4.1 - Double trouble. Suppose you know a right triangle...Ch. 4.1 - K-ple trouble. Suppose you have a right triangle...Ch. 4.1 - Padding around. You have a rectangular patio with...Ch. 4.1 - Pythagoras goes the distance. Plot the points (5,...Ch. 4.1 - Ahoy there! (H) Your exotic sailboat, which you...Ch. 4.2 - Standing guard. Draw the floor plan of a gallery...Ch. 4.2 - Art appreciation. State the Art Gallery Theorem.Ch. 4.2 - Upping the ante. How many guards do you need for a...Ch. 4.2 - Keep it safe. At what vertices would you place...Ch. 4.2 - Puttoing guards in their place. For each floor...Ch. 4.2 - Guarding the Guggenheim. The Art Gallery Theorem...Ch. 4.2 - TriangulatIng the Louvre (H). Triangulate the...Ch. 4.2 - Triangulating the Clark. Triangulate the floor...Ch. 4.2 - Tricolor me (ExH). For each triangulation, color...Ch. 4.2 - Tricolor hue. For each triangulation, color the...Ch. 4.2 - One-third. Write the number 6 as a sum of three...Ch. 4.2 - Easy watch. Draw a floor plan of a museum with six...Ch. 4.2 - Two watches (S). Draw the floor plan of a museum...Ch. 4.2 - Mirror, mirror on the wall. Consider the floor...Ch. 4.2 - Nine needs three (H). Draw a floor plan for a...Ch. 4.2 - One-third again (ExH). If a natural number is...Ch. 4.2 - Square museum (S). If a museum has only...Ch. 4.2 - Worst squares (H). Draw examples of museums with...Ch. 4.2 - Pie are squared. The circumference of a circle of...Ch. 4.2 - I can see the light. Suppose you are in a...Ch. 4.2 - Less than. Youve tnangulated your polygon and...Ch. 4.2 - Greater than. Youve triangulated your polygon and...Ch. 4.2 - Counting the colors. Your polygon has 40 vertices....Ch. 4.2 - Only red. Twelve of your polygons vertices have...Ch. 4.2 - Totaling triangles. If a polygon has n sides, it...Ch. 4.3 - Defining gold. Explain what makes a rectangle a...Ch. 4.3 - Approximating gold. Which of these numbers is...Ch. 4.3 - Approximating again. Which of the following...Ch. 4.3 - Same solution. Why does the equation l1=1l have...Ch. 4.3 - X marks the unkonw (ExH). Solve eachh equation for...Ch. 4.3 - A cold tall one? Can a Golden Rectangle have a...Ch. 4.3 - Fold the gold (H). Suppose you have a Golden...Ch. 4.3 - Sheets of gold. Suppose you have two sheets of...Ch. 4.3 - Circular logic? (H). Take a Golden Rectangle and...Ch. 4.3 - Growing gold (H). Take a Golden Rectangle and...Ch. 4.3 - Counterfeit gold? Draw a rectangle with its longer...Ch. 4.3 - In the grid (S). Consider the 1010 grid at left....Ch. 4.3 - A nest of gold. Consider the figure of infinitely...Ch. 4.3 - Comparing areas (ExH). Let G be a Golden Rectangle...Ch. 4.3 - Do we get gold? Lets make a rectangle somewhat...Ch. 4.3 - Do we get gold this time? (S) We now describe...Ch. 4.3 - A silver lining? (H) Consider the diagonal in the...Ch. 4.3 - Prob. 20MSCh. 4.3 - Going platinum. Determine the dimensions of a...Ch. 4.3 - Golden triangles. Draw a right triangle with one...Ch. 4.3 - Prob. 23MSCh. 4.3 - Prob. 24MSCh. 4.3 - Prob. 25MSCh. 4.3 - Power beyond the mathematics. Provide several...Ch. 4.3 - Special K. As a student at the University of...Ch. 4.3 - Special x. Find all values of x satisfying the...Ch. 4.3 - In search of x. Solve each equation for x:...Ch. 4.3 - Adding a square. Your school Healthy Eating garden...Ch. 4.3 - Golden Pythagoras (H). If you have a Golden...Ch. 4.4 - To tile or not to tile. Which of the following...Ch. 4.4 - Shifting Into symmetry. Shown below are small...Ch. 4.4 - Prob. 3MSCh. 4.4 - Prob. 4MSCh. 4.4 - Symmetric scaling (ExH). Each of the two patterns...Ch. 4.4 - Build a super. Draw a 1,2,5 right triangle in the...Ch. 4.4 - Another angle. Look at the 5-unit super-tile you...Ch. 4.4 - Super-super. Surround your 5-unit super-tile with...Ch. 4.4 - Expand forever (H). If you continue the process of...Ch. 4.4 - Prob. 10MSCh. 4.4 - Expand again. Take your 4.unit equilateral...Ch. 4.4 - One-answer supers. Here is a Pinwheel Pattern. For...Ch. 4.4 - Prob. 14MSCh. 4.4 - Many answer supers (H). Shown here are pictures of...Ch. 4.4 - Fill er up? (ExH) For each tile below, could...Ch. 4.4 - Prob. 18MSCh. 4.4 - Prob. 19MSCh. 4.4 - Prob. 20MSCh. 4.4 - Penrose tiles. Roger Penrose constructed two tiles...Ch. 4.4 - Expand forever. Why does any shape that can be...Ch. 4.4 - Super total. Recall that the Pinwheel Triangle has...Ch. 4.4 - Prob. 26MSCh. 4.4 - XY-tiles. The trapezoidal tile on the left has one...Ch. 4.4 - School spirit. Your dorm bathroom is tiled using...Ch. 4.4 - T-total (H). Suppose you start with one small...Ch. 4.5 - Its nice to be regular. What makes a polygon a...Ch. 4.5 - Keeping it Platonic. What makes a solid a regular...Ch. 4.5 - Countem up. How many faces, edges, and vertices...Ch. 4.5 - Defending duality. Explain why the cube and the...Ch. 4.5 - The eye of the beholder. Suppose you have models...Ch. 4.5 - Drawing solids. Draw each solid by completing the...Ch. 4.5 - Count. For each of the regular solids, take the...Ch. 4.5 - Soccer counts (ExH). Look at a soccer ball. Take...Ch. 4.5 - A solid slice (S). For each regular solid, imagine...Ch. 4.5 - Siding on the cube. Suppose we start with the...Ch. 4.5 - Cube slices (H). Consider slicing the cube with a...Ch. 4.5 - Dual quads (S). Suppose you have a cube with edges...Ch. 4.5 - Super dual. Suppose you take a cube with edges of...Ch. 4.5 - Self-duals. Suppose you have a tetrahedron having...Ch. 4.5 - Not quite regular (ExH). Suppose you allow...Ch. 4.5 - Truncated solids. Slice off all the vertices of...Ch. 4.5 - Stellated solids. Take each regular solid and...Ch. 4.5 - Prob. 24MSCh. 4.5 - Here we celeb rate the power of algebra as a...Ch. 4.5 - Here we celeb rate the power of algebra as a...Ch. 4.5 - Here we celeb rate the power of algebra as a...Ch. 4.5 - Here we celeb rate the power of algebra as a...Ch. 4.5 - Here we celeb rate the power of algebra as a...Ch. 4.6 - Walkind the walk. Here are three walks from corner...Ch. 4.6 - Missing angle in action. The triangles below are...Ch. 4.6 - Slippery X. A triangle is drawn on a sphere. Can...Ch. 4.6 - A triangular trio. The sphere below has three...Ch. 4.6 - Saddle sores. The triangle at right is drawn on a...Ch. 4.6 - Travel agent. In each of the following three...Ch. 4.6 - Travel agent. In each of the following three...Ch. 4.6 - Travel agent. In each of the following three...Ch. 4.6 - Latitude losers (H). In each of the following...Ch. 4.6 - Latitude losers (H). In each of the following...Ch. 4.6 - Latitude losers (H). In each of the following...Ch. 4.6 - Spider and bug. For each pair of points on the...Ch. 4.6 - Spider and bug. For each pair of points on the...Ch. 4.6 - Spider and bug. For each pair of points on the...Ch. 4.6 - Spider and bug. For each pair of points on the...Ch. 4.6 - Spider and bug. For each pair of points on the...Ch. 4.6 - Big angles (H). What is the largest value we can...Ch. 4.6 - Many angles (S). Draw three different great...Ch. 4.6 - Quads in a plane. Measure the sum of the angles of...Ch. 4.6 - Quads on the sphere. Below are quadrilaterals on...Ch. 4.6 - Parallel lines (ExH). On a plane, if you draw a...Ch. 4.6 - Cubical spheres (ExH). Take a cube. Put a point in...Ch. 4.6 - Tetrahedral spheres. Lets do a similar calculation...Ch. 4.6 - Dodecahedral spheres. This Mindscape is the same...Ch. 4.6 - Total excess. Using the observations from the...Ch. 4.6 - What is the sum of the three angles? Why? Consider...Ch. 4.6 - What is the sum of the angles of your triangle? Is...Ch. 4.6 - Removing a slice of the pie. Complete the...Ch. 4.6 - Conjuring up a conjecture. Make a conjecture about...Ch. 4.6 - Tetrahedral angles. What is the sum of the angles...Ch. 4.6 - Here we celebrate the power of algebra as a...Ch. 4.6 - Here we celebrate the power of algebra as a...Ch. 4.6 - Here we celebrate the power of algebra as a...Ch. 4.6 - Here we celebrate the power of algebra as a...Ch. 4.6 - Here we celebrate the power of algebra as a...Ch. 4.7 - At one with the univers. Below is a sketch of a...Ch. 4.7 - Are we there yet? Why does the information x=4 not...Ch. 4.7 - Plain places. Plot the following points in the...Ch. 4.7 - Big stack. If you take a huge number of sheets of...Ch. 4.7 - A bigger stack. If you take a huge number of...Ch. 4.7 - On the level in two dimensions. Pictured in the...Ch. 4.7 - On the level in two dimensions (S). Pictured in...Ch. 4.7 - On the level in four dimensions. Pictured in the...Ch. 4.7 - Tearible 2s. In the pictures below, describe how...Ch. 4.7 - Dare not to tear? For the figures in the Tearible...Ch. 4.7 - Unlinking (H). Using the fourth dimension,...Ch. 4.7 - Unknotting. Describe how you would unknot the...Ch. 4.7 - Prob. 13MSCh. 4.7 - Edgy hypercubes (H). Produce drawings of the...Ch. 4.7 - Prob. 15MSCh. 4.7 - Prob. 16MSCh. 4.7 - Doughnuts in dimensions. Suppose we have a...Ch. 4.7 - Assembly required (S). As promised in the...Ch. 4.7 - Slicing the cube. Take a 3-dimensional cube...Ch. 4.7 - Here we celebrate the power of algebra as a...Ch. 4.7 - Here we celebrate the power of algebra as a...Ch. 4.7 - Here we celebrate the power of algebra as a...Ch. 4.7 - Here we celebrate the power of algebra as a...Ch. 4.7 - Here we celebrate the power of algebra as a...
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