Leaving Cert Maths Samples

_________________________________________ leavingcertsolutions.com ________________________________________ 2015 Question 8 – Ordinary Paper 1 Solution (a) Given 2 ( ) 96 0.03 f x p x x = = − Number of barrels x 500 1000 1500 2000 2500 Daily profit p 40,500 66,000 76,500 72,000 52,500 Using the table function on the Casio, ( )83 7 f x G Method 1: mode table (3), f ( x ) = 96alpha x − 0.03alpha x 2 start 500, end 25000, step 500 = (or just find f (500), f (1000), etc) ( 10 marks ) (b) Daily profit Number of barrels ( 10 marks ) (c) (i) To find the daily profit when 1750 barrels are traded, go to 1750 on the x axis Draw a line parallel to the y axis, where it cutes the graph, read the value of y 76,125 y = (you should check your answer by finding f (1750) ( 5 marks ) (ii) To find the number of barrels traded when the profit is 60,000, go to 60,000 on the y axis, draw a line parallel the the x axis, where it cuts the graph, read the values of x 850 or 2350 x x = = ( 10 marks ) (d) (i) We know 2 96 0.03 p x x = − To find the maximum profit, find 96 0.06 dp x dx = − 96 96 0.06 0 0.06 96 1600 0.06 x x x − = ⇒ − = − ⇒ = = ( 5 marks ) (ii) Now find the maximum profit: 2 96(1600) 0.03(1600) €76,800 = − = ( 5 marks ) (e) Profit = 2 0.96 0.03 x x − . We want to find x if the profit is 0 Solve ( ) 2 96 0.03 0 0.03 3200 0 0, 3200 x x x x x x − = ⇒ − − = ⇒ = = If he sells more than 3200 l/day, he will make no profit ( k > 3200 l/day) ( 5 marks ) Comment: Very long question. Most ordinary level students would have no hope. p = 60,000