Date | November 2020 | Marks available | 3 | Reference code | 20N.2.SL.TZ0.T_5 |
Level | Standard Level | Paper | Paper 2 | Time zone | Time zone 0 |
Command term | Show that | 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, .
, , , and . Angle and angle . 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 .
Part of this sequence is shown in the table.
At the end of the hour, the total volume of fuel in the tank was .
Find , the height of the tank.
Show that the volume of the tank is , correct to three significant figures.
Write down the common difference, .
Find the amount of fuel pumped into the tank in the hour.
Find the value of such that .
Write down the number of hours that the pump was pumping fuel into the tank.
Find the total amount of fuel pumped into the tank in the first hours.
Show that the tank will never be completely filled using this pump.
Markscheme
* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.
OR (M1)
Note: Award (M1) for correct substitutions in trig ratio.
OR
(M1)
Note: Award (M1) for correct substitutions in Pythagoras’ theorem.
(A1)(G2)
[2 marks]
(M1)(M1)
Note: Award (M1) for their correctly substituted area of trapezium formula, provided all substitutions are positive. Award (M1) for multiplying by . Follow through from part (a).
OR
(M1)(M1)
Note: Award (M1) for the addition of correct areas for two triangles and one rectangle. Award (M1) for multiplying by . Follow through from part (a).
OR
(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).
(A1)
(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]
(A1)
[1 mark]
(M1)
Note: Award (M1) for correct substitutions in arithmetic sequence formula.
OR
Award (M1) for a correct term seen as part of list.
(A1)(ft)(G2)
Note: Follow through from part (c) for their value of .
[2 marks]
(M1)
Note: Award (M1) for their correct substitution into arithmetic sequence formula, equated to zero.
(A1)(ft)(G2)
Note: Follow through from part (c). Award at most (M1)(A0) if their is not a positive integer.
[2 marks]
(A1)(ft)
Note: Follow through from part (e)(i), but only if their final answer in (e)(i) is positive. If their in part (e)(i) is not an integer, award (A1)(ft) for the nearest lower integer.
[1 mark]
(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 correct terms.
(A1)(ft)(G2)
Note: Follow through from part (c). Award at most (M1)(A0) if their final answer is greater than .
[2 marks]
(M1)
Note: Award (M1) for their correct substitutions into arithmetic series formula.
(A1)(ft)(G1)
Note: Award (M1)(A1) for correctly finding , provided working is shown e.g. , . Follow through from part (c) and either their (e)(i) or (e)(ii). If and their final answer is greater than , award at most (M1)(A1)(ft)(R0). If , there is no maximum, award at most (M1)(A0)(R0). Award no marks if their number of terms is not a positive integer.
(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 seen, award at most (G1)(R1)(AG). Working must be seen to follow through from parts (c) and (e)(i) or (e)(ii).
OR
(M1)
Note: Award (M1) for their correct substitution into arithmetic series formula, with .
Maximum of this function (A1)
Note: Follow through from part (c). Award at most (M1)(A1)(ft)(R0) if their final answer is greater than . Award at most (M1)(A0)(R0) if their common difference is not . Award at most (M1)(A0)(R0) if is not explicitly identified as the maximum of the function.
(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.
OR
sketch with concave down curve and labelled 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 or equivalent (A1)
Note: Award (A1) for a written explanation interpreting the sketch. Accept a comparison of values, e.g , where is the graphical maximum. Award at most (M1)(A0)(R0) if their common difference is not .
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.
OR
(M1)
Note: Award (M1) for their correctly substituted arithmetic series formula equated to .
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 or with a statement "no solution". Follow through from their part (b).
[3 marks]