IB Questionbank

User interface language: English | Español

Date	May 2022	Marks available	3	Reference code	22M.2.AHL.TZ1.2
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Find	Question number	2	Adapted from	N/A

Question

A sector of a circle, centre $O$ and radius $4.5 m$ , is shown in the following diagram.

A square field with side $8 m$ has a goat tied to a post in the centre by a rope such that the goat can reach all parts of the field up to $4.5 m$ from the post.

^{[Source: mynamepong, n.d. Goat [image online] Available at: https://thenounproject.com/term/goat/1761571/}
^{This file is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0)}
^{https://creativecommons.org/licenses/by-sa/3.0/deed.en [Accessed 22 April 2010] Source adapted.]}

Let $V$ be the volume of grass eaten by the goat, in cubic metres, and $t$ be the length of time, in hours, that the goat has been in the field.

The goat eats grass at the rate of $\frac{d V}{d t} = 0.3 t e^{- t}$ .

Find the angle $AÔB$ .

[3]

a.i.

Find the area of the shaded segment.

[5]

a.ii.

Find the area of the field that can be reached by the goat.

[4]

Find the value of $t$ at which the goat is eating grass at the greatest rate.

[2]

The goat is tied in the field for $8$ hours.

Find the total volume of grass eaten by the goat during this time.

[3]

Markscheme

$(\frac{1}{2} AÔB =) arccos (\frac{4}{4.5}) = 27.266 \dots$ (M1)(A1)

$AÔB = 54.532 \dots \approx 54.5 °$ ( $0.951764 \dots \approx 0.952$ radians) A1

Note: Other methods may be seen; award (M1)(A1) for use of a correct trigonometric method to find an appropriate angle and then A1 for the correct answer.

[3 marks]

a.i.

finding area of triangle

EITHER

area of triangle $= \frac{1}{2} \times 4 . 5^{2} \times \sin (54.532 \dots)$ (M1)

Note: Award M1 for correct substitution into formula.

$= 8.24621 \dots \approx 8.25 m^{2}$ (A1)

$AB = 2 \times \sqrt{4 . 5^{2} - 4^{2}} = 4.1231 \dots$ (M1)

area triangle $= \frac{4.1231 \dots \times 4}{2}$

$= 8.24621 \dots \approx 8.25 m^{2}$ (A1)

finding area of sector

EITHER

area of sector $= \frac{54.532 \dots}{360} \times π \times 4 . 5^{2}$ (M1)

$= 9.63661 \dots \approx 9.64 m^{2}$ (A1)

area of sector $= \frac{1}{2} \times 0.9517641 \dots \times 4 . 5^{2}$ (M1)

$= 9.63661 \dots \approx 9.64 m^{2}$ (A1)

THEN

area of segment $= 9.63661 \dots - 8.24621 \dots$

$= 1.39 m^{2} (1.39040 \dots)$ A1

[5 marks]

a.ii.

METHOD 1

$π \times 4 . 5^{2} (63.6172 \dots)$ (A1)

$4 \times 1.39040 . . . (5.56160)$ (A1)

subtraction of four segments from area of circle (M1)

$= 58.1 m^{2} (58.055 \dots)$ A1

METHOD 2

angle of sector $= 90 - 54.532 \dots (\frac{π}{2} - 0.951764 \dots)$ (A1)

area of sector $= \frac{90 - 54.532 \dots}{360} \times π \times 4 . 5^{2} (= 6.26771 \dots)$ (A1)

area is made up of four triangles and four sectors (M1)

total area $= (4 \times 8.2462 \dots) + (4 \times 6.26771 \dots)$

$= 58.1 m^{2} (58.055 \dots)$ A1

[4 marks]

sketch of $\frac{d V}{d t}$ OR $\frac{d V}{d t} = 0.110363 \dots$ OR attempt to find where $\frac{d^{2} V}{d t^{2}} = 0$ (M1)

$t = 1$ hour A1

[2 marks]

recognizing $V = \int \frac{d V}{d t} d t$ (M1)

$\int_{0}^{8} 0.3 t e^{- 1} d t$ (A1)

volume eaten is $0.299 \dots m^{3} (0.299094 \dots)$ A1

[3 marks]

Examiners report

Generally, this question was answered well but provided a good example of final marks being lost due to premature rounding. Some candidates gave a correct three significant figure intermediate answer of 27.3˚ for the angle in the right-angles triangle and then doubled it to get 54.6˚ as a final answer. This did not receive the final answer mark as the correct answer is 54.5˚ to three significant figures. Premature rounding needs to be avoided in all questions.

a.i.

[N/A]

a.ii.

Unfortunately, many candidates failed to see the connection to part (a). Indeed, the most common answer was to assume the goat could eat all the grass in a circle of radius 4.5m.

Most candidates completed this question successfully by graphing the function. A few tried to differentiate the function again and, in some cases, also managed to obtain the correct answer.

This was a question that was pleasingly answered correctly by many candidates who recognized that integration was needed to find the answer. As in part (c) a few tried to do the integration ‘by hand’, and were largely unsuccessful.

Syllabus sections

Topic 5—Calculus » SL 5.5—Integration introduction, areas between curve and x axis

Show 85 related questions

22M.2.AHL.TZ1.6d.ii:
Find the area of the shaded region on the diagram.
18M.2.SL.TZ2.T_1a.i:
Write down the value of a.
17N.2.SL.TZ0.T_4c:
Copy and complete the tree diagram.
19M.2.SL.TZ2.T_1a:
Write down the null hypothesis, H₀, for this test.
22M.1.SL.TZ2.6b.ii:
Calculate the area of this region.
18M.2.SL.TZ1.S_4a:
Write down the coordinates of the vertex of the graph of g.
SPM.1.SL.TZ0.10c:
The three points A(0, 0) , B(3, 10) and C( $a$ , 0) define the vertices of a triangle.

Find the value of $a$ , the $x$ -coordinate of C, such that the area of the triangle is equal to the area of region R.
18M.1.SL.TZ1.S_5a:
Find $\int {{{\left( {f\left( x \right)} \right)}^2}{\text{d}}x}$ .
17N.2.SL.TZ0.T_4f:
Find the probability that the tested adult is allergic to nuts given that the liquid turned blue.
18M.2.SL.TZ2.T_1a.ii:
Write down the value of b.
21M.1.SL.TZ1.13b:
Sieun observes that when the angle is $40 °$ , the ball will travel a horizontal distance of $205.5 m$ .

Find an expression for the function $l (θ)$ .
EXN.2.SL.TZ0.3b.ii:
Hence write down the domain of $h$ .
22M.1.SL.TZ2.6b.i:
Write down an integral for the area of the shaded region in diagram 2.
22M.1.SL.TZ2.6c:
Hence, determine the area enclosed between $y = 6 - x$ and $y = 1.5 x^{2} - 2.5 x + 3$ .
22M.1.SL.TZ2.6a:
Calculate the area of the shaded region in diagram 1.
SPM.1.SL.TZ0.10a:
Write down an integral for the area of region R.
22M.1.SL.TZ1.12b:
One year after the company was founded, the profit was $4$ thousand dollars.

Find an expression for $P (t)$ , when $t \geq 0$ .
22M.1.SL.TZ1.6b:
Find the exact area of the shaded region in the painting.
22M.1.SL.TZ1.6c:
Find the area of the unshaded region in the painting.
22M.2.AHL.TZ1.6d.i:
Find the area enclosed by $y = f (x)$ , the $x$ -axis and the line $x = 0.5$ .
21N.1.SL.TZ0.13a.iii:
Hence find the equation of the quadratic curve.
21N.1.SL.TZ0.13a.ii:
Hence form two equations in terms of $a$ and $b$ .
16N.2.SL.TZ0.T_6b:
Express this volume in ${\text{c}}{{\text{m}}^3}$ .
17M.2.SL.TZ2.S_8b.i:
Write down the coordinates of A.
19M.2.SL.TZ2.T_1b:
State the number of degrees of freedom.
19M.2.SL.TZ2.T_1c.i:
the expected frequency of female students who chose to take the Chinese class.
19M.2.SL.TZ2.T_1d:
State whether or not H₀ should be rejected. Justify your statement.
18M.2.SL.TZ1.S_4b:
On the grid above, sketch the graph of g for −2 ≤ x ≤ 4.
17M.2.SL.TZ2.S_8b.ii:
Write down the rate of change of $f$ at A.
17N.2.SL.TZ0.T_4d:
Find the probability that this adult is allergic to nuts and the liquid turns blue.
SPM.1.SL.TZ0.10b:
Find the area of region R.
19M.1.SL.TZ1.T_12c:
Write down the probability that the second spin is yellow, given that the first spin is blue.
17N.2.SL.TZ0.T_4g:
Estimate the number of employees, from this 38, who are allergic to nuts.
17M.2.SL.TZ2.S_8c.i:
Find the coordinates of B.
18M.3.AHL.TZ0.Hca_3c:
Hence write down a lower bound for $\sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}}$ .
19M.2.SL.TZ2.T_1e.iii:
Find the probability that at least one of the two students is female.
19M.1.SL.TZ1.T_12a:
Find the probability that both spins are yellow.
16N.2.SL.TZ0.S_6b:
The empty barrel is being filled with water. The volume $V{\text{ }}{{\text{m}}^3}$ of water in the barrel after $t$ minutes is given by $V = 0.8(1 - {{\text{e}}^{ - 0.1t}})$ . How long will it take for the barrel to be half-full?
18M.2.SL.TZ2.S_3b:
The region enclosed by the graph of $f$ , the y-axis and the x-axis is rotated 360° about the x-axis.

Find the volume of the solid formed.
19M.2.SL.TZ2.S_2b:
The region enclosed by the graph of $f$ , the $x$ -axis and the $y$ -axis is rotated 360º about the $x$ -axis. Find the volume of the solid formed.
19M.1.SL.TZ1.T_12b:
Find the probability that at least one of the spins is yellow.
19M.2.SL.TZ2.T_1e.i:
Find the probability that the student does not take the Spanish class.
17N.2.SL.TZ0.T_4a:
Find the probability that this person is not allergic to nuts.
18M.2.SL.TZ1.S_4c:
Find the area of the region enclosed by the graphs of f and g.
17N.2.SL.TZ0.S_5a:
Find the value of $p$ .
17N.2.SL.TZ0.T_4b:
Find the probability that both people chosen are not allergic to nuts.
19M.2.SL.TZ2.S_2a:
Find the $x$ -intercept of the graph of $f$ .
18M.2.SL.TZ2.T_1b.i:
Use the tree diagram to find the probability that an employee encountered traffic and was late for work.
18N.2.SL.TZ0.S_10c:
When $t$ = 0, the volume of water in the container is 2.3 m³. It is known that the container is never completely full of water during the 4 hour period.

Find the minimum volume of empty space in the container during the 4 hour period.
19M.2.SL.TZ2.T_1e.ii:
Find the probability that neither of the two students take the Spanish class.
18M.3.AHL.TZ0.Hca_3d:
Find an upper bound for $\sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}}$ .
17M.2.SL.TZ2.S_8d:
Let $R$ be the region enclosed by the graph of $f$ , the $x$ -axis, the line $x = b$ and the line $x = a$ . The region $R$ is rotated 360° about the $x$ -axis. Find the volume of the solid formed.
17M.2.SL.TZ2.S_8a:
Find the value of $p$ .
EXN.1.AHL.TZ0.10a.ii:
State in context what this value represents.
EXN.2.SL.TZ0.3c:
Find the range of $h$ .
18N.2.SL.TZ0.S_10a:
Find the volume of the container.
EXN.1.AHL.TZ0.10a.i:
Find $\int_{0}^{5} \frac{1000}{2 + t} d t$ .
18M.1.SL.TZ1.S_5b:
Part of the graph of f is shown in the following diagram.

The shaded region R is enclosed by the graph of f, the x-axis, and the lines x = 1 and x = 9 . Find the volume of the solid formed when R is revolved 360° about the x-axis.
17N.2.SL.TZ0.T_4e:
Find the probability that the liquid turns blue.
18M.3.AHL.TZ0.Hca_3a:
Find the value of $\int\limits_4^\infty {\frac{1}{{{x^3}}}{\text{d}}x}$ .
18M.2.SL.TZ2.S_3a:
Find the x-intercept of the graph of $f$ .
18M.1.SL.TZ2.S_2a:
Find $\int {\left( {6{x^2} - 3x} \right){\text{d}}x}$ .
18N.2.SL.TZ0.S_10b.ii:
During the interval $p$ < $t$ < $q$ , he volume of water in the container increases by $k$ m³. Find the value of $k$ .
17N.2.SL.TZ0.S_5b:
The following diagram shows part of the graph of $f$ .

The region enclosed by the graph of $f$ , the $x$ -axis and the lines $x = - p$ and $x = p$ is rotated 360° about the $x$ -axis. Find the volume of the solid formed.
21M.1.SL.TZ2.13a:
Find an expression for $P$ in terms of $x$ .
18M.3.AHL.TZ0.Hca_3b:
Illustrate graphically the inequality $\sum\limits_{n = 5}^\infty {\frac{1}{{{n^3}}}} < \int\limits_4^\infty {\frac{1}{{{x^3}}}{\text{d}}x} < \sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}}$ .
18M.1.SL.TZ2.T_7c:
Two girls are selected at random.

Calculate the probability that one girl answered questions in Mandarin and the other answered questions in Hindi.
21M.1.SL.TZ1.13a:
Determine whether the graph of $l$ against $θ$ is increasing or decreasing at $θ = 50 °$ .
21M.2.SL.TZ1.5a.i:
Find $\frac{d y}{d x}$ .
18M.1.SL.TZ2.S_2b:
Find the area of the region enclosed by the graph of $f$ , the x-axis and the lines x = 1 and x = 2 .
16N.2.SL.TZ0.S_6a:
Use the model to find the volume of the barrel.
21M.2.AHL.TZ1.2d.ii:
Hence find the cross-sectional area of the tunnel.
18N.2.SL.TZ0.S_10b.i:
Find the value of $p$ and of $q$ .
21M.1.SL.TZ2.13b:
The company regularly increases the number of cars it produces.

Describe how their profit changes if they increase production to over $30$ cars per month and up to $50$ cars per month. Justify your answer.
17M.2.SL.TZ2.S_8c.ii:
Find the the rate of change of $f$ at B.
21M.2.SL.TZ1.5c.i:
Write down the integral which can be used to find the cross-sectional area of the tunnel.
21M.2.SL.TZ1.5c.ii:
Hence find the cross-sectional area of the tunnel.
21M.2.AHL.TZ1.2d.i:
Write down the integral which can be used to find the cross-sectional area of the tunnel.
21N.1.SL.TZ0.13b:
Find the area of the shaded region in Irina’s design.
EXN.2.SL.TZ0.3b.i:
Find the value of $t$ at which the ball hits the ground.
EXN.1.AHL.TZ0.10c:
Determine $\int_{0}^{365} P (t) d t$ and state what it represents.
EXN.1.AHL.TZ0.10b:
Find an expression for $P$ in terms of $t$ .
EXN.2.SL.TZ0.3a:
Find an expression for the height $h$ of the ball at time $t$ .
EXN.2.SL.TZ0.6g:
Use the expression found in (f) to calculate a value for $A$ .
21N.1.SL.TZ0.13a.i:
Write down the value of $c$ .

Hide related questions

Topic 5—Calculus

Question

Markscheme

Examiners report

Syllabus sections

View options