Geom Series - Applied

	1. An author is writing a fascinating book on "The History of the Application of Sequences & Series to the Physical World". On the first day, she writes 54 pages. On the second day, she writes 36 pages and on subsequent days, she writes 2/3 of the number of pages written on the previous day. (i) How many pages does she write on the 5th day? (ii) How many pages has she written altogether on the first 5 days? (iii) What will be the maximum number of pages the book will contain? (iv) How many pages would you like to read of this book? Answers.(i) 10.6 pages. (ii) 140.7 pages. (iii) 162 pages. (iv) See Solutions for discussion!!
	2. The toll on a new road tunnel is allowed, under the construction contract, to rise by 4% at the beginning of each quarter. At the beginning of the first quarter, on 1 March 2019, the toll was set at $3. The tunnel is expected to open on 4 January 2020. (i) What will the toll be at the time of opening the tunnel (to the nearest cent)? (ii) What will the toll be, under this contact, on 4 January 2024? (ignore the outcome of the parliamentary election in May 2023). Answers.(i) $3.37. (ii) $6.32.
	3. James is trying to fill the 1,200 litre fish tank in his backyard by carring buckets of water from the tap. He does not want to do too much work each day so: on the first day, he carries 100 litres. on the second day, he carries 90 litres; on the third day, he carries 81 litres. If he continues to carry water in this way each day: (i) how long will it take James to reach the 600 litre minimum water level required to support six large goldfish? (ii) will James ever fill the fish tank? Answers.(i) 9 days. (ii) No - he gets to 1000 litres.
	4. Scott is a talented high jumper. His coach wants Scott to improve his performance through meeting a series of goals. In the first competition of the season, Scott jumps his best height to date at 1.5 m. The coach now sets Scott the goal of increasing the height of his jump by 2% at each monthly competition at which he competes from now on. (i) How high will Scott jump if he follows his Coach's instructions at the fifth monthly competition? (ii) Will Scott ever be able to reach his dream goal of 2 metres? Answers.(i) 1.62 m. (ii) Yes at the 16th competition.
	5. Alyssa is considering two job offers. JOB 1: Starting salary of $30,000 and 1% pa. JOB 2: Starting salary of $20,000 and 5% pa. (i) What is the difference between Alyssa's salary level after 6 years under the two job offers (to the nearest $10)? (ii) At the beginning of which year will the annual salary received from JOB 2 exceed the annual salary received from JOB 2? (iii) What is the difference in the gross income from each job after 13 years (to the nearest $1000)? Answer:(i) Job 1 has a better salary after 6 years by $6,000. (ii) At the beginning of the 13th year. (iii) Job 1 by $60,000.
Limiting sum	6. A ball is dropped from a height of 12 metres. On the first rebound, it rises to a height of 10 metres. in subsequent rebounds it rises to a height equal to 5/6 of that which it previously attained. (i) Draw a picture of what is happening here showing the first three bounces. (ii) Calcuate the total distance through which the ball moves before coming to rest. Answer:(ii) 132 m.
	7. The triangle ABC has a right angle at C, ∠ BAC = θ and AB = 4 units. A second right-angled triangle ACD is constructed on AC with ∠DAC = θ . A third right-angled triangle ADE is constructed on AD, with ∠DAE = θ . This process continues indefinitely, as indicated by the continuation in the diagram. (i) Find the length of AC; (ii) Hence show that the length of the interval AE is 4cos³ θ . (iii) Show that the limiting sum AC + AE + AG + ... is given by 4cot θ cosec θ .