8.3 Perimeter and Area 481 equal to the diameter of the circle. Since the diameter of a circle is twice the radius, the height of the trapezoid is 2 4 ⋅ in., or 8 in. h b b r Area of the trapezoid 1 2 ( ) 1 2 8(20 9) 4(29) 116 in. Area of the circle (4) 16 50.27 in. Area of the shaded region Area of the trapezoid Area of the circle 116 50.27 65.73 in. 1 2 2 2 2 2 2 π π π = + = ⋅ + = = = = = ⋅ ≈ = − ≈ − ≈ Thus, the area of the shaded region is approximately 65.73 in. .2 7 Now try Exercise 29 Profile in Mathematics Pierre de Fermat Pierre de Fermat (1601–1665) is considered one of the greatest mathematicians of the seventeenth century. Fermat (pronounced FairMAH), however, was a lawyer by trade and considered mathematics a hobby. Although he is mostly known for his famous “last theorem” (see the Did You Know? on page 480), Fermat also made many contributions to mathematics through his correspondence with leading mathematicians of Europe. Although Fermat published very little himself, his letters to other mathematicians influenced virtually all branches of mathematical study in his time, including analytic geometry, calculus, probability, and number theory. Example 5 Applying Lawn Fertilizer Altay plans to fertilize his lawn. The shapes and dimensions of his lot, house, driveway, pool, and rose garden are shown in Fig. 8.34. One bag of fertilizer costs $29.95 and covers 5000 ft .2 Determine how many bags of fertilizer Altay needs and the total cost of the fertilizer. Assume only full bags may be purchased. 180 ft 150 ft 40 ft 30 ft 20 ft 40 ft 16 ft 24 ft Driveway House Rose Garden 60 ft Pool Figure 8.34 Solution The total area of the lot is 150 180, ⋅ or 27,000 ft .2 To determine the area to be fertilized, subtract the area of house, driveway, pool, and rose garden from the total area. = ⋅ = = ⋅ = = ⋅ = Area of house 60 40 2400 ft Area of driveway 40 16 640 ft Area of pool 20 30 600 ft 2 2 2 The diameter of the rose garden is 24 ft, so its radius is 12 ft. r Area of rose garden (12) (144) 452.39 ft 2 2 2 π π π = = = ≈ The total area of the house, driveway, pool, and rose garden is approximately 2400 + + + 640 600 452.39, or 4092.39 ft .2 The area to be fertilized is approximately − 27,000 4092.39 ft ,2 or 22,907.61 ft .2 The number of bags of
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