Date | November 2020 | Marks available | 2 | Reference code | 20N.2.SL.TZ0.T_4 |
Level | Standard Level | Paper | Paper 2 | Time zone | Time zone 0 |
Command term | Find | Question number | T_4 | Adapted from | N/A |
Question
Hyungmin designs a concrete bird bath. The bird bath is supported by a pedestal. This is shown in the diagram.
The interior of the bird bath is in the shape of a cone with radius , height and a constant slant height of .
Let be the volume of the bird bath.
Hyungmin wants the bird bath to have maximum volume.
Write down an equation in and that shows this information.
Show that .
Find .
Using your answer to part (c), find the value of for which is a maximum.
Find the maximum volume of the bird bath.
To prevent leaks, a sealant is applied to the interior surface of the bird bath.
Find the surface area to be covered by the sealant, given that the bird bath has maximum volume.
Markscheme
* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.
(or equivalent) (A1)
Note: Accept equivalent expressions such as or . Award (A0) for a final answer of or , or any further incorrect working.
[1 mark]
OR (M1)
Note: Award (M1) for correct substitution in the volume of cone formula.
(AG)
Note: The final line must be seen, with no incorrect working, for the (M1) to be awarded.
[1 mark]
(A1)(A1)
Note: Award (A1) for , (A1) for . Award at most (A1)(A0) if extra terms are seen. Award (A0) for the term .
[2 marks]
(M1)
Note: Award (M1) for equating their derivative to zero. Follow through from part (c).
OR
sketch of (M1)
Note: Award (M1) for a labelled sketch of with the curve/axes correctly labelled or the -intercept explicitly indicated.
(A1)(ft)
Note: An unsupported is awarded no marks. Graphing the function is not an acceptable method and (M0)(A0) should be awarded. Follow through from part (c). Given the restraints of the question, is not possible.
[2 marks]
(M1)
OR
(M1)
Note: Award (M1) for substituting their in the volume formula.
(A1)(ft)(G2)
Note: Follow through from part (d).
[2 marks]
(A1)(ft)(M1)
Note: Award (A1) for their correct radius seen .
Award (M1) for correctly substituted curved surface area formula for a cone.
(A1)(ft)(G2)
Note: Follow through from parts (a) and (d).
[3 marks]