91Ó°ÊÓ

The formula used to determine the radius of the yaw mark arc is derived from a geometric relationship about two intersecting chords in a circle. In the figure, chords \(\overline{A B}\) and \(\overline{C D}\) intersect at point \(E\) in the circle. The product of the two segment lengths making up chord \(\overline{A B}, A E \times E B,\) is equal to the product of the two segment lengths making up chord \(C D, C E \times E D .\) In the next figure, the yaw mark is darkened and it is continued to form a complete circle. A chord is drawn connecting two points on the yaw mark. The middle ordinate is also drawn. The length of the middle ordinate is \(M\) and the length of the chord is \(C D .\) The middle ordinate cuts the chord into two equal pieces with each half of the chord \(\frac{C D}{2}\) units in length. The radius of the circle has length \(r\) as shown in the diagram. Applying the property to the two intersecting chords in this diagram, you get \(A E \times E B=C E \times E D .\) a. From the diagram, \(C E=\frac{C D}{2}, E D=\frac{C D}{2},\) and \(E B=M .\) You need to determine the length of the segment AE. Notice that \(A B=2 r .\) It is a diameter, which equals the length of two radii.) Also notice that \(A E=A B-E B .\) Write an algebraic expression that represents the length of \(A E .\) b. Write the algebraic expression for the product of the segments of a chord that applies to this situation. Do not simplify. c. Simplify the side of the equation that represents the product of the segments of chord \(\overline{C D} .\) Write the new equation. d. Solve the equation for \(r\) by isolating the variable \(r\) on one side of the equation. Show your work. Compare your answer with the radius formula.

A new car sells for \(27,300. It exponentially depreciates at a rate of 6.1% to \)22,100. How long did it take for the car to depreciate to this amount? Round your answer to the nearest tenth of a year.

Amber bought a used car valued at \(16,000. When this car was new, it was sold for \)28,000. If the car depreciates exponentially at a rate of 9% per year, approximately how old is the car?

Create a list of numbers whose mean, median, and mode are all 10.

Jazmine’s car originally sold for \(46,600. It depreciates exponentially at a rate of 10.3% per year. Jazmine put \)12,000 down and pays $800 per month to pay off the balance. After how many years will her car value equal the amount she paid to date for the car? What will that value be?