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Date	May 2021	Marks available	1	Reference code	21M.2.SL.TZ2.5
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 2
Command term	Explain	Question number	5	Adapted from	N/A

Question

A hollow chocolate box is manufactured in the form of a right prism with a regular hexagonal base. The height of the prism is $h cm$ , and the top and base of the prism have sides of length $x cm$ .

Given that $\sin 60 ° = \frac{\sqrt{3}}{2}$ , show that the area of the base of the box is equal to $\frac{3 \sqrt{3} x^{2}}{2}$ .

[2]

Given that the total external surface area of the box is $1200 {cm}^{2}$ , show that the volume of the box may be expressed as $V = 300 \sqrt{3} x - \frac{9}{4} x^{3}$ .

[5]

Sketch the graph of $V = 300 \sqrt{3} x - \frac{9}{4} x^{3}$ , for $0 \leq x \leq 16$ .

[2]

Find an expression for $\frac{d V}{d x}$ .

[2]

Find the value of $x$ which maximizes the volume of the box.

[2]

Hence, or otherwise, find the maximum possible volume of the box.

[2]

The box will contain spherical chocolates. The production manager assumes that they can calculate the exact number of chocolates in each box by dividing the volume of the box by the volume of a single chocolate and then rounding down to the nearest integer.

Explain why the production manager is incorrect.

[1]

Markscheme

evidence of splitting diagram into equilateral triangles M1

area $= 6 (\frac{1}{2} x^{2} \sin 60 °)$ A1

$= \frac{3 \sqrt{3} x^{2}}{2}$ AG

Note: The AG line must be seen for the final A1 to be awarded.

[2 marks]

total surface area of prism $1200 = 2 (3 x^{2} \frac{\sqrt{3}}{2}) + 6 x h$ M1A1

Note: Award M1 for expressing total surface areas as a sum of areas of rectangles and hexagons, and A1 for a correctly substituted formula, equated to $1200$ .

$h = \frac{400 - \sqrt{3} x^{2}}{2 x}$ A1

volume of prism $= \frac{3 \sqrt{3}}{2} x^{2} \times h$ (M1)

$= \frac{3 \sqrt{3}}{2} x^{2} (\frac{400 - \sqrt{3} x^{2}}{2 x})$ A1

$= 300 \sqrt{3} x - \frac{9}{4} x^{3}$ AG

Note: The AG line must be seen for the final A1 to be awarded.

[5 marks]

A1A1

Note: Award A1 for correct shape, A1 for roots in correct place with some indication of scale (indicated by a labelled point).

[2 marks]

$\frac{d V}{d x} = 300 \sqrt{3} - \frac{27}{4} x^{2}$ A1A1

Note: Award A1 for a correct term.

[2 marks]

from the graph of $V$ or $\frac{d V}{d x}$ OR solving $\frac{d V}{d x} = 0$ (M1)

$x = 8.88 (8.877382 \dots)$ A1

[2 marks]

from the graph of $V$ OR substituting their value for $x$ into $V$ (M1)

$V_{max} = 3040 {cm}^{3} (3039.34 \dots)$ A1

[2 marks]

EITHER
wasted space / spheres do not pack densely (tesselate) A1

OR
the model uses exterior values / assumes infinite thinness of materials and hence the modelled volume is not the true volume A1

[1 mark]

Examiners report

[N/A]

Syllabus sections

Topic 5—Calculus » SL 5.6—Stationary points, local max and min

Show 140 related questions

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22M.2.SL.TZ2.5b:
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17M.1.AHL.TZ2.H_10a.i:
Find the area of the window in terms of P and $r$ .
18M.2.SL.TZ1.T_4e:
Sketch the graph of y = f (x) for 0 < x ≤ 6 and −30 ≤ y ≤ 60.
Clearly indicate the minimum point P and the x-intercepts on your graph.
18M.1.SL.TZ2.T_13b:
Find the value of s.
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Calculate the area of cloth, in cm², needed to make Haruka’s bag.
18N.2.SL.TZ0.T_6e:
Use your answers to parts (c) and (d) to show that

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17M.2.SL.TZ1.T_6d.i:
Find $g’( - 1)$ .
22M.2.SL.TZ1.4c:
Find the value of $t$ at which the goat is eating grass at the greatest rate.
22M.2.AHL.TZ2.6b:
Find the maximum height reached by the ball.
EXN.1.AHL.TZ0.15b:
Find in terms of $a$ the value of $x$ at which the maximum occurs.
21N.1.AHL.TZ0.8b:
Juri starts at a height of $60$ metres and finishes at $F$ , where $x = f$ .

Sketch a possible diagram of the hill on the following pair of coordinate axes.
EXN.1.AHL.TZ0.15a:
Find $f' (x)$ .
22M.1.AHL.TZ1.17:
A function $f$ is of the form $f (t) = p e^{q \cos (r t)}, p, q, r \in ℝ^{+}$ . Part of the graph of $f$ is shown.

The points $A$ and $B$ have coordinates $A (0, 6.5)$ and $B (5.2, 0.2)$ , and lie on $f$ .

The point $A$ is a local maximum and the point $B$ is a local minimum.

Find the value of $p$ , of $q$ and of $r$ .
22M.2.AHL.TZ1.2c:
Find the value of $t$ at which the goat is eating grass at the greatest rate.
21N.1.SL.TZ0.12b.i:
Solve $S' (x) = 0$ .
21N.1.AHL.TZ0.15b.i:
Find an expression for $V$ in terms of $θ$ .
21N.1.AHL.TZ0.15b.ii:
Find the expression $\frac{d V}{d θ}$ .
21N.1.AHL.TZ0.15b.iii:
Solve algebraically $\frac{d V}{d θ} = 0$ to find the value of $θ$ that will maximize the volume, $V$ .
SPM.2.SL.TZ0.4d:
Find $P\left( x \right)$ .
17M.1.SL.TZ1.S_6a.i:
Write down the gradient of the curve of $f$ at P.
17M.2.SL.TZ2.S_8b.i:
Write down the coordinates of A.
19M.2.SL.TZ1.S_4a:
Sketch the graph of ${f''}$ on the grid below:
19M.2.SL.TZ2.T_5f:
Find the equation of the tangent line to the graph of $y = f\left( x \right)$ at $x = 2$ . Give the equation in the form $ax + by + d = 0$ where, $a$ , $b$ , and $d \in \mathbb{Z}$ .
19M.2.SL.TZ2.S_5a:
Find the population of fish at $t$ = 10.
18M.2.SL.TZ1.T_4a:
Find the value of k.
18M.1.SL.TZ2.T_13c:
Find the number of shirts produced when the cost of production is lowest.
18N.2.SL.TZ0.T_6b:
Calculate the volume, in cm³, of the bag.
17M.2.SL.TZ2.T_6f:
Write down the number of possible solutions to the equation $f(x) = 5$ .
17M.1.AHL.TZ2.H_10a.ii:
Find the width of the window in terms of P when the area is a maximum, justifying that this is a maximum.
17M.1.AHL.TZ2.H_10b:
Show that in this case the height of the rectangle is equal to the radius of the semicircle.
18M.2.SL.TZ2.T_6f:
Given that y = 2x³ − 9x² + 12x + 2 = k has three solutions, find the possible values of k.
18M.1.SL.TZ2.S_9c:
Given that there is a minimum value for C, find this minimum value in terms of $\pi$ .
18M.1.SL.TZ2.T_13a:
Find the cost of producing 70 shirts.
17M.2.SL.TZ2.S_8b.ii:
Write down the rate of change of $f$ at A.
18N.2.SL.TZ0.T_6h:
The cloth used to make Nanako’s bag costs 4 Japanese Yen (JPY) per cm².

Find the cost of the cloth used to make Nanako’s bag.
18M.2.SL.TZ2.T_6a:
Sketch the curve for −1 < x < 3 and −2 < y < 12.
19M.1.AHL.TZ1.H_8d:
Sketch the curve $y = f\left( x \right)$ , 0 ≤ $x$ ≤ 5 indicating clearly the coordinates of the maximum and minimum points and any intercepts with the coordinate axes.
SPM.2.SL.TZ0.4a:
Calculate the surface area of the box in cm².
19N.2.SL.TZ0.T_6c:
Given the design constraint that $l = \frac{150 - 2 π r^{2}}{π r}$ , show that $V = 150 r - \frac{2 π r^{3}}{3}$ .
19M.2.SL.TZ1.S_4b:
Find the $x$ -coordinates of the points of inflexion of the graph of $f$ .
17M.2.SL.TZ2.T_6d.i:
Write down the $x$ -coordinates of these two points;
18M.2.SL.TZ2.T_6b:
A teacher asks her students to make some observations about the curve.

Three students responded.
Nadia said “The x-intercept of the curve is between −1 and zero”.
Rick said “The curve is decreasing when x < 1 ”.
Paula said “The gradient of the curve is less than zero between x = 1 and x = 2 ”.

State the name of the student who made an incorrect observation.
17M.2.SL.TZ2.T_6a:
Write down the $y$ -intercept of the graph.
17M.2.SL.TZ1.T_6d.ii:
Hence justify that $g$ is decreasing at $x = - 1$ .
19M.2.SL.TZ2.S_5b:
Find the rate at which the population of fish is increasing at $t$ = 10.
19N.2.SL.TZ0.T_6e:
Using your answer to part (d), show that $V$ is a maximum when $r$ is equal to $\sqrt{\frac{75}{π}} cm$ .
19N.2.SL.TZ0.T_6h:
Use your answer to part (f) to identify the shape of the speaker with the best quality of sound.
17M.2.SL.TZ2.T_6c.ii:
Find $f(2)$ .
18N.2.SL.TZ0.T_6f:
Find $\frac{{{\text{d}}A}}{{{\text{d}}x}}$ .
17M.2.SL.TZ2.S_8c.i:
Find the coordinates of B.
18M.2.SL.TZ1.T_4b:
Using your value of k , find f ′(x).
19M.2.SL.TZ1.S_4c:
Hence find the values of $x$ for which the graph of $f$ is concave-down.
19N.2.SL.TZ0.T_6g:
Calculate the maximum value of $V$ .
18N.1.SL.TZ0.S_10b.ii:
Hence, find the equation of L in terms of $a$ .
18N.1.SL.TZ0.S_10a:
Find the coordinates of P.
17N.1.SL.TZ0.S_7:
Consider $f(x) = \log k(6x - 3{x^2})$ , for $0 < x < 2$ , where $k > 0$ .

The equation $f(x) = 2$ has exactly one solution. Find the value of $k$ .
16N.2.SL.TZ0.S_10a:
(i) Find the value of $c$ .

(ii) Show that $b = \frac{\pi }{6}$ .

(iii) Find the value of $a$ .
18N.2.SL.TZ0.T_6d:
Write down and simplify an expression in x and y for the area of cloth, A, used to make Nanako’s bag.
17M.1.SL.TZ1.S_6b:
Determine the concavity of the graph of $f$ when $4 < x < 5$ and justify your answer.
19M.1.SL.TZ2.T_15a:
Find the value of $P$ if no vases are sold.
19N.2.SL.TZ0.T_6f:
Find the length of the cylinder for which $V$ is a maximum.
19M.1.AHL.TZ2.H_8b:
Given that there is one inscribed cone having a maximum volume, show that the volume of this cone is $\frac{{32\pi {R^3}}}{{81}}$ .
18N.2.SL.TZ0.T_6g:
Use your answer to part (f) to show that the width of Nanako’s bag is 12 cm.
17M.2.SL.TZ2.T_6c.i:
Show that $a = 8$ .
19N.2.SL.TZ0.T_6d:
Find $\frac{d V}{d r}$ .
18N.1.SL.TZ0.S_10b.i:
Find $f'\left( x \right)$ .
17M.1.SL.TZ1.S_6a.ii:
Find the equation of the normal to the curve of $f$ at P.
18N.1.SL.TZ0.S_10c:
The graph of $f$ has a local minimum at the point Q. The line L passes through Q.

Find the value of $a$ .
19N.2.SL.TZ0.T_6a:
Write down an expression for $V$ , the volume (cm³) of the speaker, in terms of $r$ , $l$ and $π$ .
19M.2.SL.TZ1.T_6a:
Show that $k = - 6$ .
19M.1.AHL.TZ1.H_8a:
Write down the $x$ -coordinate of the point of inflexion on the graph of $y = f\left( x \right)$ .
17M.2.SL.TZ1.T_6b.ii:
Find the equation of the tangent to the graph of $y = g(x)$ at $x = 2$ . Give your answer in the form $y = mx + c$ .
19M.1.SL.TZ2.T_15b:
Differentiate $P\left( x \right)$ .
18M.1.AHL.TZ1.H_9b.i:
Show that there is exactly one point of inflexion, B, on the graph of $y = f\left( x \right)$ .
18M.1.AHL.TZ1.H_9a:
The graph of $y = f\left( x \right)$ has a local maximum at A. Find the coordinates of A.
17M.2.SL.TZ2.T_6d.ii:
Write down the intervals where the gradient of the graph of $y = f(x)$ is positive.
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.
18M.1.AHL.TZ2.H_4:
Consider the curve $y = \frac{1}{{1 - x}} + \frac{4}{{x - 4}}$ .

Find the x-coordinates of the points on the curve where the gradient is zero.
16N.2.SL.TZ0.T_6f:
Using your answer to part (e), find the value of $r$ which minimizes $A$ .
18N.2.SL.TZ0.T_6c:
Use this value to write down, and simplify, the equation in x and y for the volume of Nanako’s bag.
18M.2.AHL.TZ1.H_9a:
Show that there are exactly two points on the curve where the gradient is zero.
17M.2.SL.TZ2.T_6g:
The equation $f(x) = m$ , where $m \in \mathbb{R}$ , has four solutions. Find the possible values of $m$ .
16N.2.SL.TZ0.S_10b:
(i) Write down the value of $k$ .

(ii) Find $g(x)$ .
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.
16N.2.SL.TZ0.T_6e:
Find $\frac{{{\text{d}}A}}{{{\text{d}}r}}$ .
17M.2.SL.TZ2.S_8a:
Find the value of $p$ .
18M.1.SL.TZ2.S_9b:
Show that $C = 20\pi {r^2} + \frac{{320\pi }}{r}$ .
19M.2.SL.TZ2.S_5c:
Find the value of $t$ for which the population of fish is increasing most rapidly.
18M.2.SL.TZ2.T_6d:
Find $\frac{{{\text{dy}}}}{{{\text{dx}}}}$ .
18N.2.SL.TZ0.S_10a:
Find the volume of the container.
17M.2.SL.TZ2.T_6b:
Find $f'(x)$ .
SPM.2.SL.TZ0.4e:
Find the least number of boxes which must be sold each week in order to make a profit.
18M.1.AHL.TZ1.H_9b.ii:
The coordinates of B can be expressed in the form B $\left( {{2^a},\,b \times {2^{ - 3a}}} \right)$ where a, b $\in \mathbb{Q}$ . Find the value of a and the value of b.
17M.2.SL.TZ1.T_6b.i:
Show that $k = 6$ .
17M.2.SL.TZ1.T_6a:
Find $g'(x)$ .
18M.2.SL.TZ1.T_4c:
Use your answer to part (b) to show that the minimum value of f(x) is −22 .
18M.2.AHL.TZ1.H_9d:
The shaded region is rotated by 2 $\pi$ about the $y$ -axis. Find the volume of the solid formed.
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 $k$ .
19M.2.SL.TZ2.T_5d:
Find $f'\left( x \right)$ .
17M.2.SL.TZ2.T_6e:
Write down the range of $f(x)$ .
19M.2.SL.TZ1.T_6b:
Find the coordinates of the local minimum.
SPM.2.SL.TZ0.4b:
Calculate the length AG.
19M.1.AHL.TZ1.H_8b:
find the value of $f\left( 1 \right)$ .
17M.2.SL.TZ1.T_6e:
Find the $y$ -coordinate of the local minimum.
16N.2.SL.TZ0.T_6c:
Write down, in terms of $r$ and $h$ , an equation for the volume of this water container.
16N.2.SL.TZ0.T_6a:
Write down a formula for $A$ , the surface area to be coated.
16N.2.SL.TZ0.T_6h:
Find the least number of cans of water-resistant material that will coat the area in part (g).
19M.2.SL.TZ1.T_6d:
Determine the equation of the normal at $x = - 2$ in the form $y = mx + c$ .
17M.2.SL.TZ1.T_6c:
Use your answer to part (a) and the value of $k$ , to find the $x$ -coordinates of the stationary points of the graph of $y = g(x)$ .
21M.2.SL.TZ1.5a.ii:
Hence find the maximum height of the tunnel.
21M.2.AHL.TZ1.2a.ii:
Hence find the maximum height of the tunnel.
18N.2.SL.TZ0.S_10b.i:
Find the value of $p$ and of $q$ .
16N.2.SL.TZ0.T_6d:
Show that $A = \pi {r^2} + \frac{{1\,000\,000}}{r}$ .
SPM.2.SL.TZ0.4c:
Find the number of boxes that should be sold each week to maximize the profit.
16N.2.SL.TZ0.S_10c:
(i) Find $w$ .

(ii) Hence or otherwise, find the maximum positive rate of change of $g$ .
18M.2.AHL.TZ1.H_9c:
The normal at P cuts the curve again at the point Q. Find the $x$ -coordinate of Q.
19N.2.SL.TZ0.T_6b:
Write down an equation for the surface area of the speaker in terms of $r$ , $l$ and $π$ .
19M.2.SL.TZ1.T_6c:
Write down the interval where the gradient of the graph of $f\left( x \right)$ is negative.
17N.1.SL.TZ0.T_11c:
Find the value of $a$ and of $b$ .
21M.2.SL.TZ2.5f:
Hence, or otherwise, find the maximum possible volume of the box.
16N.2.SL.TZ0.T_6g:
Find the value of this minimum area.
18M.2.AHL.TZ1.H_9b:
Find the equation of the normal to the curve at the point P.
19M.1.AHL.TZ1.H_8c:
find the value of $f\left( 4 \right)$ .
18M.1.SL.TZ2.S_9a:
Express h in terms of r.
17M.2.SL.TZ2.S_8c.ii:
Find the the rate of change of $f$ at B.
19M.1.AHL.TZ2.H_8a:
Show that the volume of the cone may be expressed by $V = \frac{\pi }{3}\left( {2R{h^2} - {h^3}} \right)$ .
EXN.2.SL.TZ0.2e:
Hence or otherwise find the minimum length of ribbon required.
19M.2.SL.TZ2.T_5e:
Find the gradient of the graph of $y = f\left( x \right)$ at $x = 2$ .
21N.1.AHL.TZ0.15a:
Show that $r = \frac{6}{2 + θ}$ .
EXN.1.AHL.TZ0.15c:
Hence find the value of $a$ for which $y$ has the smallest possible maximum value.
EXN.2.SL.TZ0.2a:
Find an expression for the total length of the ribbon $L$ in terms of $x$ and $y$ .
EXN.2.SL.TZ0.2b:
Show that $L = 2 x + \frac{300}{x} + 22$
EXN.2.SL.TZ0.2c:
Find $\frac{d L}{d x}$
EXN.2.SL.TZ0.2d:
Solve $\frac{d L}{d x} = 0$
EXN.2.SL.TZ0.6d:
Use differentiation to show that $12 a + 4 b + c = 0$ .
21N.1.SL.TZ0.12a:
Find $S' (x)$ .
21N.1.SL.TZ0.12b.ii:
Interpret your answer to (b)(i) in context.
21N.1.AHL.TZ0.8a.i:
Identify the $x$ value of the point where $| h' (x) |$ has its maximum value.
21N.1.AHL.TZ0.8a.ii:
Interpret this point in the given context.

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Topic 5—Calculus

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