SECTION 1.2 The Distance and Midpoint Formulas 19 (a) Find an estimate for the desired intersection point. (b) Find the distance from ( ) 1.4, 1.3 to the midpoint found in part (a). 64. Net Sales The figure illustrates the growth in the net sales of Amazon.com’s e-service from 2018 through 2022. Use the midpoint formula to estimate the net sales of Amazon.com’s e-service in 2020. How does your result compare to the reported value of $386.06 billion? Source: Amazon 2022 Annual Report. 2018 0 200 400 600 Amazon e-services Net Sales ($ billions) 2019 513.98 232.89 2020 Year 2021 2022 65. Student Loan Debt In 2008, the average student loan debt for a university graduate was $21,211. In 2022, the average student loan debt for a university graduate was $29,417. Use the midpoint formula to estimate the average student loan debt for a university graduate in 2015. How does your result compare to the actual student loan debt for a university graduate of $29,289? Source: Statista.com 66. Challenge Problem Geometry Verify that the points ( ) 0, 0 , ( ) a, 0 , and ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ a a 2 , 3 2 are the vertices of an equilateral triangle. Then show that the midpoints of the three sides are the vertices of a second equilateral triangle. 67. Challenge Problem Geometry Find the midpoint of each diagonal of a square with side of length s. Draw the conclusion that the diagonals of a square intersect at their midpoints. [Hint: Use ( ) ( ) ( ) ( ) s s s s 0, 0 , 0, , , 0 , and , as the vertices of the square.] 68. Challenge Problem Geometry A point P is equidistant from ( ) −5, 1 and ( ) − 4, 4 . Find the coordinates of P if its y-coordinate is twice its x-coordinate. 69. Challenge Problem Geometry For any parallelogram, prove that the sum of the squares of the lengths of the sides equals the sum of the squares of the lengths of the diagonals. [Hint: Use ( ) 0, 0 , ( ) a, 0 , ( ) +a b c , , and ( ) b c , as the vertices of the parallelogram.] (b) If the right fielder is located at ( ) 310, 15 how far is it from the right fielder to second base? (c) If the center fielder is located at ( ) 300, 300 , how far is it from the center fielder to third base? 60. Little League Baseball Refer to Problem 58. Overlay a rectangular coordinate system on a Little League baseball diamond so that the origin is at home plate, the positive x-axis lies in the direction from home plate to first base, and the positive y-axis lies in the direction from home plate to third base. (a) What are the coordinates of first base, second base, and third base? Use feet as the unit of measurement. (b) If the right fielder is located at ( ) 180, 20 , how far is it from the right fielder to second base? (c) If the center fielder is located at ( ) 220, 220 , how far is it from the center fielder to third base? 61. Distance between Moving Objects A Ford Focus and a Freightliner Cascadia truck leave an intersection at the same time. The Focus heads east at an average speed of 60 miles per hour, while the Cascadia heads south at an average speed of 45 miles per hour. Find an expression for their distance apart d (in miles) at the end of t hours. 62. Distance of a Moving Object from a Fixed Point A hot-air balloon, headed due east at an average speed of 15 miles per hour and at a constant altitude of 100 feet, passes over an intersection (see the figure). Find an expression for the distance d (measured in feet) from the balloon to the intersection t seconds later. 15 mph East 100 ft 63. Drafting Error When a drafter draws three lines that are to intersect at one point, the lines may not intersect as intended and subsequently will form an error triangle. If this error triangle is long and thin, one estimate for the location of the desired point is the midpoint of the shortest side. The figure shows one such error triangle. x y (1.4, 1.3) (2.7, 1.7) (2.6, 1.5) 1.7 1.5 1.3 2.6 2.7 1.4 ‘Are You Prepared?’ Answers 1. 8 2. 5 3. + = + = = 11 60 121 3600 3721 61 2 2 2 4. ( ) 0, 0 5. bh 1 2 6. True
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