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Date	November 2020	Marks available	2	Reference code	20N.2.SL.TZ0.T_5
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	T_5	Adapted from	N/A

Question

A large underground tank is constructed at Mills Airport to store fuel. The tank is in the shape of an isosceles trapezoidal prism, $ABCDEFGH$ .

$AB = 70 m$ , $AF = 200 m$ , $AD = 40 m$ , $BC = 40 m$ and $CD = 110 m$ . Angle $ADC = 60 °$ and angle $BCD = 60 °$ . The tank is illustrated below.

Once construction was complete, a fuel pump was used to pump fuel into the empty tank. The amount of fuel pumped into the tank by this pump each hour decreases as an arithmetic sequence with terms $u_{1}, u_{2}, u_{3}, \dots, u_{n}$ .

Part of this sequence is shown in the table.

At the end of the $2 nd$ hour, the total volume of fuel in the tank was $88 200 m^{3}$ .

Find $h$ , the height of the tank.

[2]

Show that the volume of the tank is $624 000 m^{3}$ , correct to three significant figures.

[3]

Write down the common difference, $d$ .

[1]

Find the amount of fuel pumped into the tank in the $13th$ hour.

[2]

Find the value of $n$ such that $u_{n} = 0$ .

[2]

e.i.

Write down the number of hours that the pump was pumping fuel into the tank.

[1]

e.ii.

Find the total amount of fuel pumped into the tank in the first $8$ hours.

[2]

Show that the tank will never be completely filled using this pump.

[3]

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$\sin 60 ° = \frac{h}{40}$ OR $\tan 60 ° = \frac{h}{20}$ (M1)

Note: Award (M1) for correct substitutions in trig ratio.

$20^{2} + h^{2} = 40^{2} (\sqrt{40^{2} - 20^{2}})$ (M1)

Note: Award (M1) for correct substitutions in Pythagoras’ theorem.

$(h =) 34.6 (m) (\sqrt{1200}, 20 \sqrt{3}, 34.6410 \dots)$ (A1)(G2)

[2 marks]

$\frac{1}{2} (70 + 110) (34.6410 \dots) \times 200$ (M1)(M1)

Note: Award (M1) for their correctly substituted area of trapezium formula, provided all substitutions are positive. Award (M1) for multiplying by $200$ . Follow through from part (a).

$(2 \times \frac{1}{2} \times 20 \times 34.6410 \dots + 70 \times 34.6410 \dots) \times 200$ (M1)(M1)

Note: Award (M1) for the addition of correct areas for two triangles and one rectangle. Award (M1) for multiplying by $200$ . Follow through from part (a).

$70 \times 34.6410 \dots \times 200 + 2 \times \frac{1}{2} \times 34.6410 \dots \times 20 \times 200$ (M1)(M1)

Note: Award (M1) for their correct substitution in volume of cuboid formula. Award (M1) for correctly substituted volume of triangular prism(s). Follow through from part (a).

$623538 \dots$ (A1)

$624000 (m^{3})$ (AG)

Note: Both an unrounded answer that rounds to the given answer and the rounded value must be seen for the (A1) to be awarded.

[3 marks]

$(d =) - 1800$ (A1)

[1 mark]

$(u_{13} =) 45000 + (13 - 1) (- 1800)$ (M1)

Note: Award (M1) for correct substitutions in arithmetic sequence formula.
OR
Award (M1) for a correct $4^{th}$ term seen as part of list.

$23400 (m^{3})$ (A1)(ft)(G2)

Note: Follow through from part (c) for their value of $d$ .

[2 marks]

$0 = 45000 + (n - 1) (- 1800)$ (M1)

Note: Award (M1) for their correct substitution into arithmetic sequence formula, equated to zero.

$(n =) 26$ (A1)(ft)(G2)

Note: Follow through from part (c). Award at most (M1)(A0) if their $n$ is not a positive integer.

[2 marks]

e.i.

$25$ (A1)(ft)

Note: Follow through from part (e)(i), but only if their final answer in (e)(i) is positive. If their $n$ in part (e)(i) is not an integer, award (A1)(ft) for the nearest lower integer.

[1 mark]

e.ii.

$(S_{8} =) \frac{8}{2} (2 \times 45000 + (8 - 1) \times (- 1800))$ (M1)

Note: Award (M1) for their correct substitutions in arithmetic series formula. If a list method is used, award (M1) for the addition of their $8$ correct terms.

$310 000 (m^{3}) (309 600)$ (A1)(ft)(G2)

Note: Follow through from part (c). Award at most (M1)(A0) if their final answer is greater than $624 000$ .

[2 marks]

$(S_{25} =) \frac{25}{2} (2 \times 45000 + (25 - 1) \times (- 1800)), (S_{25} =) \frac{25}{2} (45000 + 1800)$ (M1)

Note: Award (M1) for their correct substitutions into arithmetic series formula.

$S_{25} = 585000 (m^{3})$ (A1)(ft)(G1)

Note: Award (M1)(A1) for correctly finding $S_{26} = 585000 (m^{3})$ , provided working is shown e.g. $(S_{26} =) \frac{26}{2} (2 \times 45000 + (26 - 1) \times (- 1800))$ , $(S_{26} =) \frac{26}{2} (45000 + 0)$ . Follow through from part (c) and either their (e)(i) or (e)(ii). If $d < 0$ and their final answer is greater than $624 000$ , award at most (M1)(A1)(ft)(R0). If $d > 0$ , there is no maximum, award at most (M1)(A0)(R0). Award no marks if their number of terms is not a positive integer.

$585000 (m^{3}) < 624000 (m^{3})$ (R1)

Hence it will never be filled (AG)

Note: The (AG) line must be seen. If it is omitted do not award the final (R1). Do not follow through within the part.
For unsupported $(S_{25}) = 585000$ seen, award at most (G1)(R1)(AG). Working must be seen to follow through from parts (c) and (e)(i) or (e)(ii).

$(S_{n} =) \frac{n}{2} (2 \times 45000 + (n - 1) \times (- 1800))$ (M1)

Note: Award (M1) for their correct substitution into arithmetic series formula, with $n$ .

Maximum of this function $585225 (m^{3})$ (A1)

Note: Follow through from part (c). Award at most (M1)(A1)(ft)(R0) if their final answer is greater than $624 000$ . Award at most (M1)(A0)(R0) if their common difference is not $- 1800$ . Award at most (M1)(A0)(R0) if $585 225$ is not explicitly identified as the maximum of the function.

$585225 (m^{3}) < 624000 (m^{3})$ (R1)

Hence it will never be filled (AG)

Note: The (AG) line must be seen. If it is omitted do not award the final (R1). Do not follow through within the part.

sketch with concave down curve and labelled $624000$ horizontal line (M1)

Note: Accept a label of “tank volume” instead of a numerical value. Award (M0) if the line and the curve intersect.

curve explicitly labelled as $(S_{n} =) \frac{n}{2} (2 \times 45000 + (n - 1) \times (- 1800))$ or equivalent (A1)

Note: Award (A1) for a written explanation interpreting the sketch. Accept a comparison of values, e.g $585225 (m^{3}) < 624000 (m^{3})$ , where $585225$ is the graphical maximum. Award at most (M1)(A0)(R0) if their common difference is not $- 1800$ .

the line and the curve do not intersect (R1)

hence it will never be filled (AG)

Note: The (AG) line must be seen. If it is omitted do not award the final (R1). Do not follow through within the part.

$624000 = \frac{n}{2} (2 \times 45000 + (n - 1) \times (- 1800))$ (M1)

Note: Award (M1) for their correctly substituted arithmetic series formula equated to $624000 (623538)$ .

Demonstrates there is no solution (A1)

Note: Award (A1) for a correct working that the discriminant is less than zero OR correct working indicating there is no real solution in the quadratic formula.

There is no (real) solution (to this equation) (R1)

hence it will never be filled (AG)

Note: At most (M1)(A0)(R0) for their correctly substituted arithmetic series formula $= 624000, 623538$ or $622800$ with a statement "no solution". Follow through from their part (b).

[3 marks]

Examiners report

[N/A]

e.i.

[N/A]

e.ii.

[N/A]

Syllabus sections

Topic 1—Number and algebra » SL 1.2—Arithmetic sequences and series

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22M.2.SL.TZ2.2a.i:
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22M.2.SL.TZ1.2a.i:
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19M.1.AHL.TZ2.H_1:
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18M.2.SL.TZ2.T_4e:
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Find the value of m.
17M.1.AHL.TZ1.H_7b:
determine the value of $\sum\limits_{r = 1}^N {{u_r}}$ .
17M.1.SL.TZ2.T_5b:
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18M.2.SL.TZ2.T_4b:
Calculate the café’s total profit for the first 12 weeks.
16N.2.AHL.TZ0.H_12a:
Find an expression for ${A_1}$ and show that ${A_2} = {1.004^2}x + 1.004x$ .
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19M.2.SL.TZ2.T_4d.iii:
Hence find the total amount of antibiotic, in mg, that Ted receives during the first $k$ days.
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19M.2.SL.TZ1.T_5a:
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17N.1.SL.TZ0.S_2c:
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16N.1.AHL.TZ0.H_6b:
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17M.1.AHL.TZ1.H_7a:
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17M.1.SL.TZ1.T_5b:
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Write down an equation, in terms of ${u_1}$ and $d$ , for the amount of the drug that she receives on the seventh day.
18N.1.SL.TZ0.S_3a:
Find u₈.
16N.1.SL.TZ0.S_9e:
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16N.2.AHL.TZ0.H_12e:
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18M.1.SL.TZ1.T_7b:
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16N.2.AHL.TZ0.H_12d:
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Write down the common difference, $d$ .
20N.2.SL.TZ0.T_5f:
Find the total amount of fuel pumped into the tank in the first $8$ hours.
20N.2.SL.TZ0.T_5g:
Show that the tank will never be completely filled using this pump.
19M.2.SL.TZ2.T_4d.ii:
The daily amount of antibiotic Ted receives will first be less than 0.06 mg on the $k$ th day. Find the value of $k$ .
19N.2.SL.TZ0.T_3e:
Each triangular pane has an area of $1.84 m^{2}$ .

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Write down the number of hours that the pump was pumping fuel into the tank.
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Write down the value of $Σ_{i = 1}^{9} u_{i}$ .
EXN.1.SL.TZ0.12b:
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Find the amount of antibiotic, in mg, that Ted receives on the fifth day.
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Find the exact value of ${S_k}$ .
17N.2.SL.TZ0.T_2e:
Calculate the total distance Carlos runs in the first year.
18M.2.AHL.TZ1.H_7a:
Show that there will be approximately 2645 fish in the lake at the start of 2020.
16N.1.SL.TZ0.S_9c:
Find $d$ , giving your answer as an integer.
19N.2.SL.TZ0.T_3c:
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16N.2.AHL.TZ0.H_12b:
(i) Write down a similar expression for ${A_3}$ and ${A_4}$ .

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18M.2.SL.TZ1.S_7b:
Hence find the value of n such that $\sum\limits_{k = 1}^n {{x_k} = 861}$ .
19M.2.SL.TZ2.T_4c:
Calculate the total amount of the drug, in mg, that she receives.
20N.2.SL.TZ0.T_5b:
Show that the volume of the tank is $624 000 m^{3}$ , correct to three significant figures.
19M.2.SL.TZ2.S_10d:
An infinite geometric series is given as ${S_\infty } = a + \frac{a}{{\sqrt 2 }} + \frac{a}{2} + \ldots$ , $a \in {\mathbb{Z}^ + }$ .

Find the largest value of $a$ such that ${S_\infty } < F$ .
18M.2.AHL.TZ1.H_7b:
Find the approximate number of fish in the lake at the start of 2042.
19M.2.SL.TZ1.S_7a:
Write down the first three non-zero terms of ${w_n}$ .
17M.1.SL.TZ2.T_5a:
Diagram $n$ is formed with 52 sticks. Find the value of $n$ .
21M.1.SL.TZ1.8a.i:
Charlie ran on day $20$ of his fitness programme.
20N.2.SL.TZ0.T_5d:
Find the amount of fuel pumped into the tank in the $13th$ hour.
19N.2.SL.TZ0.T_3a:
Find the number of triangular panes in the $12 th$ level.
SPM.1.SL.TZ0.2c:
Find the total cost of buying 2 tickets in each of the first 16 rows.
17M.1.SL.TZ1.T_5a:
Find the distance from the base of this ladder to the top rung.
19N.2.SL.TZ0.T_3d:
Find the maximum number of complete levels that Maegan can build.
18M.1.SL.TZ2.S_7b:
Let $p = {c^2}$ and $q = {c^3}$ . Find the value of $\sum\limits_{n = 1}^{20} {{u_n}}$ .
21M.2.SL.TZ2.3a.ii:
the total number of seats in the concert hall.
21M.2.SL.TZ2.3a.i:
the number of seats in the last row.
17M.1.AHL.TZ2.H_3a:
the value of $d$ ;
19M.2.SL.TZ2.T_4a.ii:
Write down an equation, in terms of ${u_1}$ and $d$ , for the amount of the drug that she receives on the eleventh day.
18M.1.AHL.TZ2.H_5b:
A particular geometric sequence has u₁ = 3 and a sum to infinity of 4.

Find the value of d.
19M.2.SL.TZ1.S_7b.ii:
Find the value of $m$ .
19M.2.SL.TZ2.S_10c:
Show that $F = 3240$ .
19M.2.SL.TZ1.T_5d:
Find the total amount John has paid to insure his bicycle for the first 5 years.
20N.2.SL.TZ0.T_5e.i:
Find the value of $n$ such that $u_{n} = 0$ .
16N.2.AHL.TZ0.H_12c:
Write down an expression for ${A_n}$ in terms of $x$ on the day before Mary turned 18 years old showing clearly the value of $n$ .
17N.2.SL.TZ0.T_2c:
Calculate the total distance, in kilometres, Rosa runs in the first 50 training sessions.
EXN.3.AHL.TZ0.2a:
Find an expression for the width of $U_{n}$ in centimetres.
19M.2.SL.TZ1.T_5b:
Find the value of the bicycle during the 5th year. Give your answer to two decimal places.
EXN.1.SL.TZ0.12c:
A game is played in which the arrow attached to the centre of the disc is spun and the sector in which the arrow stops is noted. If the arrow stops in sector $1$ the player wins $10$ points, otherwise they lose $2$ points.

Let $X$ be the number of points won

Find $E (X)$ .
19M.2.SL.TZ2.S_10a:
Find the value of $k$ .
19M.2.SL.TZ1.S_7b.i:
Find the value of $r$ .
21M.1.SL.TZ1.8a.ii:
Daniella ran on day $20$ of her fitness programme.
EXN.3.AHL.TZ0.2b:
Given the width of a pixel is approximately $0.025 cm$ , find the number of squares in the final image.
21N.2.SL.TZ0.2a:
Calculate the percentage increase in applications from the first year to the second year.
21N.2.SL.TZ0.2b.ii:
Find an expression for $u_{n}$ .
21N.2.SL.TZ0.2b.iii:
Find the number of student applications the university expects to receive when $n = 11$ . Express your answer to the nearest integer.
21N.2.SL.TZ0.2c:
Write down an expression for $v_{n}$ .
21N.2.SL.TZ0.2d:
Calculate the total amount of acceptance fees paid to the university in the first $10$ years.
21N.2.SL.TZ0.2e:
Find $k$ .

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Topic 1—Number and algebra

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