480 CHAPTER 8 Geometry Example 3 Comparing Pizzas Jana wishes to order a large cheese pizza. She can choose among three pizza parlors in town: Antonio’s, Steve’s, and Dorsey’s. Antonio’s large cheese pizza is a round 16-in.-diameter pizza that sells for $15. Steve’s large cheese pizza is a round 14-in.-diameter pizza that sells for $12. Dorsey’s large cheese pizza is a square 12-in. by 12-in. pizza that sells for $10. All three pizzas have the same thickness. To get the most for her money, from which pizza parlor should Jana order her pizza? Solution To determine the best value, we will calculate the cost per square inch of pizza for each of the three pizzas. To do so, we will divide the cost of each pizza by its area. The areas of the two round pizzas can be determined using the formula for the area of a circle, A r .2 π = Since the radius is half the diameter, we will use r 8 = and r 7 = for Antonio’s and Steve’s large pizzas, respectively, and we will use the π key on our calculator. The area for the square pizza can be determined using the formula for the area of a square, A s .2 = We will use s 12. = π π π π π π = = = ≈ = = = ≈ = = = r r s Area of Antonio’s pizza (8) (64) 201.06 in. Area of Steve’s pizza (7) (49) 153.94 in. Area of Dorsey’s pizza (12) 144 in. 2 2 2 2 2 2 2 2 2 Now, to determine the cost per square inch of pizza, we will divide the cost of the pizza by the area of the pizza. Cost per square inch of Antonio’s pizza $15 201.06 in. $0.0746 2 ≈ ≈ Thus, Antonio’s pizza costs about $0.0746, or about 7.5 cents, per square inch. Cost per square inch of Steve’s pizza $12 153.94 in. $0.0780 2 ≈ ≈ Thus, Steve’s pizza costs about $0.0780, or about 7.8 cents, per square inch. Cost per square inch of Dorsey’s pizza $10 144 in. $0.0694 2 = ≈ Thus, Dorsey’s pizza costs about $0.0694, or about 6.9 cents, per square inch. Since the cost per square inch of pizza is the lowest for Dorsey’s pizza, Jana would get the most pizza for her money by ordering her pizza from Dorsey’s. 7 Now try Exercise 51 Did You Know? Fermat’s Last Theorem m Andrew J. Wiles In 1637, Pierre de Fermat, an amateur French mathematician (see Profile in Mathematics on page 481), scribbled a note in the margin of the book Arithmetica by Diophantus. The note would haunt mathematicians for centuries. Fermat stated that the generalized form of the Pythagorean theorem, + = a b c , n n n has no positive integer solutions where ≥ n 3. Fermat’s note concluded, “I have a truly marvelous demonstration of this proposition, which this margin is too narrow to contain.” This conjecture became known as Fermat’s last theorem. A formal proof of this conjecture escaped mathematicians until on September 19, 1994, Andrew J. Wiles of Princeton University announced he had completed a proof. It took Wiles more than 8 years of work to accomplish the task. Wiles was awarded the Wolfskehl prize at Göttingen University in Germany in acknowledgment of his achievement. Example 4 Determine the Shaded Area Determine the shaded area of the following figure. Use the π key on your calculator and round your answer to the nearest hundredth. 4 in. 9 in. 20 in. Solution To determine the area of the shaded region, we will subtract the area of the circle from the area of the trapezoid. Notice that the height of the trapezoid is Charles Rex Arbogast/ AP Images
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