6.1 Extended Questions (Section B, SL)

3 marks

The following diagram shows the graph of the function $f$ defined by $f (x) = 4 x^{2} + 4 x - 4 x^{3}$ as well as the graph of $y = g (x)$ for a function $g$ , with both functions defined for all $x \in ℝ .$

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Points $C$ and $D$ are the non-negative $x$ -axis intercepts of f, while $E$ is the local maximum on the graph of $f$ . Point $F$ is the point of intersection between the graphs of $f$ and $g$ . The shaded region R represents the area between the graph of $g$ and the $x$ -axis. The shaded region $S$ represents the area above the positive $x$ -axis between the graphs of $f$ and $g$ .

Find the coordinates of $C$ , $D$ and $E$ , giving your answers correct to three significant figures.

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3 marks

The graph of $g$ is a parabola with $x$ -intercepts $(\frac{1}{2}, 0)$ and $(\frac{3}{2}, 0)$ , and vertex $(1, 2)$

Find an expression for $g (x)$ .

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3 marks

Hence find the area of region R.

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6 marks

The following diagram shows a cylindrical log with a wedge cut from it. The area of the major sector $A O B$ is equal to the area of the shaded region S. The length $l$ is equal to the distance between points E and F on the diagram above

ib1d-ai-sl-6-1-ib-maths-hard

Find the volume of the log.

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2 marks

A supermarket manager wishes to gather information about the spending habits of the store’s customers. During each of his lunchbreaks on Monday through Friday of a given week, he chooses 24 customers at random and notes the total cost, in dollars ($), of the items in their baskets when they check out at the tills. The results of his survey are represented by the following cumulative frequency graph.

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Find the median total cost of the items these customers had in their baskets.

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3 marks

Find the interquartile range of the total cost of the items these customers had in their baskets.

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3 marks

Given that two thirds of customers had a total of more than $$ p$ of goods in their baskets, find the value of $p$ .

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4 marks

The same survey information is represented by the following table:

Total cost ($m) of goods in basket	$0 < m \leq 20$	$20 < m \leq 40$	$40 < m \leq 70$	$70 < m \leq 90$
Frequency	9	$q$	$r$	20

Find the value of $q$ and the value of $r$ .

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3 marks

In an average week, the manager estimates that the store has a total of 3600 customers.

Use the results of the manager’s survey to estimate the number of customers in a week who have goods totalling more than $50 in their baskets.

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2 marks

(i)

Explain why the manager’s survey sample might not provide an accurate representation of the spending habits of all the shop’s customers.

(ii)

Suggest a sampling method that might obtain a more representative sample.

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2 marks

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.

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

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

$h (t) = 5.59 + 3.6 \cos (k π t)$

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

Use the above information to determine:

(i)

The distance of point

A

from point

C

(ii)

The height of point

C

above point

O

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3 marks

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 $C$ , determine the exact distance of point $A$ from the tip of one of the other blades of the screw.

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3 marks

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

Given that the argument of the cosine in the equation for $h (t)$ is measured in radians, use this information to determine the value of the constant $k$ .

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2 marks

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.

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3 marks

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

$\tan O \hat{Q} C = \frac{3}{2} \tan O \hat{P} C$

Determine the distance of point $Q$ from point $P$ .

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4 marks

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

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Number of hours, $h$	$2 < h \leq 7$	$7 < h \leq 9$	$9 < h \leq 15$	$15 < h \leq 21$	$21 < h \leq 24$
Frequency	4	$a$	$b$	16	$c$

DP IB Maths: AA HL

Topic Questions

6.1 Extended Questions (Section B, SL)