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QUESTION 19 (7 marks) A circular-based bowl has been positioned symmetrically on a Cartesian plane as shown in the diagram. y 4 The bowl has a shape that can be generated by rotating the curve y = S x —l about the y-axis — X for 4 <x<7.6 cm. The bowl is being filled with a liquid at the rate of 77 cm’s L. Determine the rate at which the depth of liquid is increasing when the depth of liquid reaches one-third of the height of the bowl.
QUESTION 14 (5 marks) The time ¢ (months) that it takes before a phone owner cracks the screen on their phone can be modelled by an exponentially distributed random variable ) = 0.16e~ 1 t>0 0 otherwise a) Show that f(¢) is a probability density function. [1 mark] b) Determine the probability that a phone owner cracks the screen on their phone within 1 year. [2 marks] Three-quarters of phone owners take between 1 and m months before they crack the screen on their phone. c) Determine the value of m. [2 marks]
QUESTION 17 (7 marks) The area between the graphs of the functions y =4x and y = 2x? is rotated about the y-axis to form a solid of revolution with a volume of V units. Determine the exact value of V.
QUESTION 2 Determine the area of the shaded region between the graphs of the functions y = lsec[£] and y = ZCOS(EJ as shown. 3 3 2 -3 Not to scale (A) 5.29 units? (B) 5.5 units® (C) 5.65 units” (D) 5.71 units?
QUESTION 13 (6 marks) The area under the graph of the function f(x)= 0.2¢7%** for 1< x <9 is shaded. f(x) 1 9 & Not to scale a) Use Simpson’s rule with four intervals to determine an approximation for this area. [4 marks] b) Use a calculus approach to evaluate the reasonableness of your area approximation from Question 13a). [2 marks]
QUESTION 17 (5 marks) The region between the x-axis and the curve of the function y =1+ sin(Zx) for 0 < x < Z is rotated about the x-axis to form a solid of revolution. 2 Determine the volume of this solid. Express your answer in simplest form.
QUESTION 4 The time taken for students to answer questions in a class is assumed to be a random variable X with an exponential distribution that has the probability density function f(x):{/leix x>0 0 otherwise The mean of X 1s l The mean length of time taken for students to answer questions in this class is 15 seconds. The probability that the next question in this class is answered between 8 seconds and 17 seconds is (A) 0.05 (B) 0.12 (C) 0.22 (D) 0.26
QUESTION 11 (6 marks) An aerial view of the surface of a dam 6 km in length i1s symmetrically positioned on a Cartesian plane as shown. A dam wall 1s located along the y-axis. 2 2 -2 The surrounding edge of the dam can be modelled by the ellipse (x 7 ) + % =1 for0<x<6. y Dam wall X Not to scale a) Use Simpson’s rule with four strips to determine an approximate area of the surface of the dam. [4 marks] b) Evaluate the reasonableness of this approximation. [2 marks]
QUESTION 12 (6 marks) Consider the semicircle shown below. a) Determine the equation of the semicircle in the form y = f(x). b) Determine the volume formed by rotating the semicircle about the x-axis. c) Evaluate the reasonableness of your answer to 12b). [2 marks] [2 marks] [2 marks]
QUESTION 14 (5 marks) An exponential random variable X with parameter A > 0 has a probability density function given by _ [ re™™ x>0 1) {O otherwise 1 The mean of X 1s ;\ The length of time that car owners keep their first car is an exponentially distributed random variable with a mean of 4 years. a) Determine the probability that an individual keeps their first car for between 3 and 5 years. [3 marks] 50% of all individuals keep their first car for at most k years. b) Determine the value of k. [2 marks]
QUESTION 9 The time in minutes between the arrival of customers at a certain shop is assumed to be a random variable X with an exponential distribution that has the probability density function 0.1207 0-12 x>0 0 otherwise f(x) = | A customer arrives at the shop. The probability that the next customer arrives within 30 to 60 seconds to the nearest percent is (A) 3% (B) 5% (C) 7% (D) 11%
QUESTION 11 (4 marks) The bounded region between the graphs of the functions y = —1+sec =| and y = 0. 1x” over a certain domain is shaded as shown. The two functions intersect at the origin and point A. 7 A 6 x a) Determine the coordinates of point A. [1 mark] b) Calculate the area of the shaded region. [1 mark] The shaded region is rotated about the x-axis to form a solid of revolution. c) Determine the volume of the solid formed. [2 marks]
QUESTION 14 (4 marks) At a certain location a biologist measures the width of a river to be 12 m. She also records the depth of the river at regular 2 m interval widths as shown. wien) | oo | 2 | 4 | 6 | s | wo | 2 The biologist estimates the cross-sectional area of the river at this location to be 15 m”. Use Simpson’s rule to evaluate the reasonableness of this estimation. Justify your area calculation and decision regarding reasonableness using mathematical reasoning.
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