SECTION 12.3 Geometric Sequences; Geometric Series 885 (a) What is the length of the arc of the 10th swing? (b) On which swing is the length of the arc first less than 1 foot? (c) After 15 swings, what total length has the pendulum swung? (d) When it stops, what total length has the pendulum swung? 88. Bouncing Balls A ball is dropped from a height of 30 feet. Each time it strikes the ground, it bounces up to 0.8 of the previous height. (a) What height does the ball bounce up to after it strikes the ground for the third time? (b) How high does it bounce after it strikes the ground for the nth time? (c) How many times does the ball need to strike the ground before its bounce is less than 6 inches? (d) What total vertical distance does the ball travel before it stops bouncing? 89. Retirement Christine contributes $100 each month to her 401(k). What will be the value of Christine’s 401(k) after the 360th deposit (30 years) if the per annum rate of return is assumed to be 8% compounded monthly? 90. Saving for a Home Esmeralda wants to purchase a new home. Suppose that she invests $400 per month into a mutual fund. If the per annum rate of return of the mutual fund is assumed to be 6% compounded monthly, how much will Esmeralda have for a down payment after the 36th deposit (3 years)? 91. Tax-Sheltered Annuity Don contributes $500 at the end of each quarter to a tax-sheltered annuity (TSA). What will the value of the TSA be after the 80th deposit (20 years) if the per annum rate of return is assumed to be 5% compounded quarterly? 92. Retirement Ray contributes $1000 to an individual retirement account (IRA) semiannually. What will the value of the IRA be when Ray makes his 30th deposit (after 15 years) if the per annum rate of return is assumed to be 7% compounded semiannually? 93. Sinking Fund Scott and Alice want to purchase a vacation home in 10 years and need $50,000 for a down payment. How much should they place in a savings account each month if the per annum rate of return is assumed to be 3.5% compounded monthly? 94. Sinking Fund For a child born in 2024, the cost of a 4-year college education at a public university is projected to be $185,000. Assuming a 4.75% per annum rate of return compounded monthly, how much must be contributed to a college fund every month to have $185,000 in 18 years when the child begins college? 95. Grains of Wheat on a Chess Board In an old fable, a commoner who had saved the king’s life was told he could ask the king for any just reward. Being a shrewd man, the commoner said,“A simple wish, sire. Place one grain of wheat on the first square of a chessboard, two grains on the second square, four grains on the third square, continuing until you 30' 24' 19.2' have filled the board. This is all I seek.” Compute the total number of grains needed to do this to see why the request, seemingly simple, could not be granted. (A chessboard consists of 8 8 64 × = squares.) 96. Shading Squares Look at the figure. What fraction of the square is eventually shaded if the indicated shading process continues indefinitely? 97. Multiplier Suppose that, throughout the U.S. economy, individuals spend 90% of every additional dollar that they earn. Economists would say that an individual’s marginal propensity to consume is 0.90. For example, if Jane earns an additional dollar, she will spend 0.9 1 $0.90 ( ) = of it. The individual who earns $0.90 (from Jane) will spend 90% of it, or $0.81. This process of spending continues and results in an infinite geometric series as follows: … 1, 0.90, 0.90 , 0.90 , 0.90 , 2 3 4 The sum of this infinite geometric series is called the multiplier. What is the multiplier if individuals spend 90% of every additional dollar that they earn? 98. Multiplier Refer to Problem 97. Suppose that the marginal propensity to consume throughout the U.S. economy is 0.95. What is the multiplier for the U.S. economy? 99. Stock Price One method of pricing a stock is to discount the stream of future dividends of the stock. Suppose that a stock pays $P per year in dividends, and historically, the dividend has been increased i% per year. If you desire an annual rate of return of r%, this method of pricing a stock states that the price that you should pay is the present value of an infinite stream of payments: P P i r P i r P i r Price 1 1 1 1 1 1 2 3 ( ) ( ) = + ⋅ + + + ⋅ + + + ⋅ + + + The price of the stock is the sum of an infinite geometric series. Suppose that a stock pays an annual dividend of $4.00, and historically, the dividend has been increased 3% per year.You desire an annual rate of return of 9%.What is the most you should pay for the stock? 100. Stock Price Refer to Problem 99. Suppose that a stock pays an annual dividend of $2.50, and historically, the dividend has increased 4% per year. You desire an annual rate of return of 11%.What is the most that you should pay for the stock?
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