Extended Questions (Section B, SL) (DP IB Maths: AA HL)

The diagram below shows a part of the graph of the function             

Point is the point of intersection between the graph and the y-axis. Write down the coordinates of point .

Find .

Using the graph, explain why the equation  must have exactly two distinct real solutions.

Point B is the point on the graph with -coordinate   .

Given that  ,  find the gradient of the tangent line to the graph at point .

Points  and  are the points on the graph at which the tangent lines are perpendicular to the tangent line at point .

By first determining the gradient of the tangents at points  and , find the -coordinates of points  and .

Given that point  lies between points and  on the graph, find the equation of the tangent line to the graph at point .

After escaping from a research station, a small population of rabbits has become established on an island in the Southern Ocean.  Scientists have begun to study this rabbit population, and have determined that the number of rabbits, , at a time  months after the beginning of the study can be modelled by the function

Where  is a positive constant.

Determine the number of rabbits on the island at the beginning of the study.

Show that

The model predicts that the population of rabbits will double in the first two months after the beginning of the study.

Use this information to show that  .
   

The Strike A Light! matchstick company produces matchsticks with a length,  mm, that is normally distributed with mean 45 and variance .

The probability that  is greater than 45.37 is 0.1714.

Find .

Andrew has a box of Strike A Light! matches with fifteen matchsticks remaining in it.  Those matchsticks may be assumed to be a random sample.  Let  represent the number of matchsticks in Andrew’s box with lengths less than 44.5 mm.

Find .

Find the probability that exactly one of the matchsticks in Andrew’s box has a length less than 44.5 mm.

A Strike A Light! matchstick is selected at random and is found to have a length greater than 44.5 mm.

Find the probability that the length of the matchstick is between 44.63 mm and 45.37 mm.

K.C. Jones & Company produces tunnels for model railroad layouts. Each tunnel has the form of a right prism, and the cross-section of one of the tunnels the company produces is shown in the diagram below. The upper and right-hand borders of the shaded area are parallel to the -axis and -axis respectively, and all units are in centimetres.

The shape of the opening of the tunnel may be modelled by the function

where  is a positive constant.

Points  and  are the points where the tunnel opening meets the -axis in the diagram.

Find the coordinates of points  and .

The maximum height of the tunnel opening above the -axis is 8 cm.

Use this information to determine the value of .

By setting up and solving an appropriate definite integral, show that the area of the tunnel opening is   .  You must use calculus and show the steps of your working.

The material from which the tunnel is made has a density of 1060 .

Given that the mass of the tunnel is 2067 g, find the length of the tunnel.

Badon Iron Works is building a new ship called the Gargantuan, which will be a full-sized replica of the original RMS Titanic.  Eleanor is an engineer at the company, and is involved with construction and testing of the ship’s screws (commonly known as ‘propellers’).  The diagram below depicts one of the ship’s screws mounted in the testing facility.

Point is the centre of the screw, which is fixed in place so that the screw is able to rotate about it.  Point  is the marked tip of one of the three identical blades of the screw.  Point  is the point on the horizontal floor of the testing facility that lies directly below point .  Points , , ,  and  lie at all times in the same plane.

The height,  m, of point  above the testing facility floor once the screw begins to rotate may be modelled by the function

where  is the time in seconds since the screw began rotating, and  is a constant.

Use the above information to determine:

The distance of point  from point .

The height of point  above point .

Given that the tips of the three blades of the screw are located at equal distances from each other around the circumference of a circle with centre , determine the exact distance of point  from the tip of one of the other blades of the screw.

When it is rotating, the screw makes 75 complete revolutions every minute.

Given that the argument of the cosine in the equation for is measured in radians, use this information to determine the value of the constant .

Paul, a mathematician, has been hired as a consultant on the Gargantuan project.  Because of his height of 1.96 m, Eleanor is concerned about whether he will be able to walk safely beneath the screw while it is rotating.

Determine whether Eleanor is right to be concerned, giving a mathematical reason for your answer.

The screw has been locked in place so that point  is at its highest possible position above the floor.  Paul is standing at point , which is at a distance of 9.69 m from point .  He walks towards point  until he arrives at point , which is located such that

Determine the distance of point  from point .

Given that point  remains fixed at its highest possible position above the floor, determine the area of triangle .