Problem in Real World Contexts (full paper)

1 Liao wants to cycle around Singapore. He finds the following information below about the two types of shared e-bikes that he can choose from. Bike A Cost Rates Regular $1.00 per min Family $4.00 per min Special Pass* 7-Day Pass $6.90 30-Day Pass $9.90 90-Day Pass $26.90 *Free for first 30mins, $1.00 for every subsequent 30mins. (only for regular bikes) Bike B Cost Rates Regular $1.00 for first 30 mins $0.50 for every next 10 mins Family $4.00 per 30 mins Special Pass** 7-Day Pass $5.90 30-Day Pass $9.90 90-Day Pass $24.90 **Free for first 60 mins, $0.50 for every subsequent 10 mins. (only for regular bikes) (a) Liao cycles with his family in Sembawang Park for 40 mins. He chooses the Family Bike A. He estimates that the total cost is $240. Show that Liao is correct. [1] (b) Liao buys the 7-Day Pass for Bike A. He wants to travel around the perimeter of Singapore. The distance travelled is about 150 km. He travels about 30 km/h. How much does he need to pay including the special pass? [2] (c) Liao and Abid cycle together in East Coast Park. Liao chooses Regular Bike A. He buys the 7-Day Pass. Abid chooses Regular Bike B. The distance of East Coast Park is about 15 km. Liao travels 20 km/h while Abid travels 15 km/h more than Liao. The price for the bike will keep increasing even if they are not cycling. Abid waits for Liao at the end of their journey. Abid claims that he needs to pay more because he waits for Liao. Is his claim correct? Justify your decision. Show all your workings. [7]

2 A petrol oil tanker is a motor vehicle designed to transport gasoline to petrol filling stations. Its tank is shaped like a cylindrical container, with a diameter 3 m and a length 15 m as shown in Diagram I. Diagram II shows the cross-sectional view of the tank containing a certain amount of liquid gasoline. The line AB represents the width of the horizontal surface of the liquid in contact with the cylindrical tank. To satisfy the safety storage conditions for transportation, the liquid gasoline must not exceed 95% of the tank. (a) It is given that AB = 1.8 m and ∠BMO = 90°. Show that the volume in this liquid is safe for transportation. [3] (b) Calculate the height of the liquid gasoline from line AB. [1] (c) A petrol filling station requires an underground tank of 40 000 gallons of liquid gasoline. Petrol tankers are dispatched to the station to top up once its total petrol quantity is 20% of its maximum amount. Calculate the minimum number of petrol tankers required for each top up. Justify your decision. Show all your workings. [6]

3 Zhang wanted to have a shed house of his own in his backyard as shown below. The shed house, shaped like a prism, has a length of 7 m and has four sloping roofs. It also has a door of 2 m. CD = DE = EF = FG = 1.2 m AG = BC = 1.5 m and AB = 5.1 m. CDEFG is part of a regular heptagon (7-sided). (a) Find angle DEF. [1] (b) Zhang wanted to put up a ‘Welcome’ sign board on the line DF. What is the length of the board? [2]

(c) Zhang has a budget of $2000 to spend on buying a shed house flooring. A sketch of the floor plan is shown below. All angles shown in the diagram are right angles and all dimensions are in meters. Zhang planned to use porcelain tiles to cover the areas that are not the bedroom and use wooden tiles to cover his bedroom. He found this information about the two tiles as shown below. Types of tiles Dimensions Price Porcelain 30 cm by 30 cm A box of 20 tiles costs $71.80 Wooden 20 cm by 20 cm A box of 25 tiles costs $80.20 Are both types of flooring materials within Zhang’s budget? Justify your decisions. Show all your workings. [7]

4 A factory manufactures ceramic bowls and glass cups. A ceramic bowl is made by removing the bottom part from an opened hemisphere with diameter 15 cm as shown in Figure 1a. The circular area of the bottom part has a diameter of 7 cm. The shaded part is the bottom part that should be removed. Figure 1b shows the cross-sectional view of the bowl. (a) Show that the height of a bowl is approximately 6.63 cm. [1] Bowls are arranged such that after each bowl, a piece of cardboard (black rectangles) with a thickness of about 5 mm, is placed to ensure that the bowls are in place and that they are not being scratched. The top view of how the bowls is arranged, as shown below. Each box has only one layer of bowls. (1 cm = 10 mm) Top view (b) A cuboid box measures 30 cm by 6.63 cm by 70.8 cm. How many such bowls can the box fit? [2]

(c) A glass cup, usually shaped like an opened cylinder, is used to contain beverages like soda and beers. It has a diameter of 7 cm and a height of 15 cm. Two companies suggested the following types of container used to store the cups. The information of both companies is shown below. PacknGo ServiceTime Front view Front view Number of cups: 10 Number of cups: 10 Length: 15 cm Length: 15 cm Material: Cardboard Material: Cardboard Price: $5.50 per squared cm. Price: $4.30 per squared cm. The factory wanted to pack and sell 50 cups in a day. Which company should he choose? Justify your decision. Show all your workings. [7]

5 Every National Day, decorations are put up on streets and housing estates. Flags are commonly used as decorations. The figure below shows two flags. Flag 1 is an isosceles triangular flag with AC = BC = 27 cm and angle ACB = 22°. Flag 2 is a rectangular flag by removing an equilateral triangle XRS (shaded part) from rectangle PQRS. PQ : AB = 3 : 4. Both flags have the same height. (a) In Flag 1, show that AB = 20 cm. [1] (b) Find the height of Flag 2. [2]

(c) Mr Tan, a town council manager, plans on hanging flags near a HDB housing estate. He finds the information below: The spacing between the flags is 10 cm. For every 10 cm spacing, one flag will be hung. Flags • Material used: Polyester • 1 cm2 of polyester costs $0.03 (per flag) • It is sold in area measured in whole number Strings • Material used: Nylon • 1 m of nylon costs $0.10 (sold in rolls) • It is sold in length measured in 3 significant figures HDB blocks • The standard floor-to-ceiling height is about 2.6 meters. • Top floor units have higher ceiling, about a height of 2.75 meters. The HDB housing estate block has 13 storeys from ground. The 13th storey is the top floor unit. Mr Tan, a town council, plans to hang the flags to the 13th storey such that it is 68° from ground. Which flag should he choose? Justify your decision. Show all your workings. [7]

6 Xuan is going to stay in a hotel in Kular Lumpur for a week. She needs to work out how much money, in Singapore dollars, she will need to cover her living expenses. She finds the information below to help her work out her expenses. The costs are all given in Ringgit (RM). Estimate costs in Kular Lumpur (for a week) Item Cost Hotel RM 2390 Transportation to hotel RM 230 Air ticket RM 382 Meal RM 70 per day Local transportation RM 119 per day (a) The exchange rate between Ringgit and Singapore dollars changes every month. The graph below shows the monthly exchange rate in 2023. (i) Write down the exchange rate between Singapore Dollars and Ringgit in May. Give your answer correct to 3 decimal places. [1] (ii) Xuan wants to convert her daily meal cost to dollars. Use the graph to work out the difference between the greatest and least possible daily meal costs in dollars. Give your answer correct to the nearest cents. [3] 3.27 3.27 3.33 3.32 3.37 3.44 3.44 3.41 3.43 3.46 3.48 3.5 3.15 3.2