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 and radius , is shown in the following diagram.
A square field with side 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 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 be the volume of grass eaten by the goat, in cubic metres, and be the length of time, in hours, that the goat has been in the field.
The goat eats grass at the rate of .
Find the angle .
Find the area of the shaded segment.
Find the area of the field that can be reached by the goat.
Find the value of at which the goat is eating grass at the greatest rate.
The goat is tied in the field for hours.
Find the total volume of grass eaten by the goat during this time.
Markscheme
(M1)(A1)
( 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]
finding area of triangle
EITHER
area of triangle (M1)
Note: Award M1 for correct substitution into formula.
(A1)
OR
(M1)
area triangle
(A1)
finding area of sector
EITHER
area of sector (M1)
(A1)
OR
area of sector (M1)
(A1)
THEN
area of segment
A1
[5 marks]
METHOD 1
(A1)
(A1)
subtraction of four segments from area of circle (M1)
A1
METHOD 2
angle of sector (A1)
area of sector (A1)
area is made up of four triangles and four sectors (M1)
total area
A1
[4 marks]
sketch of OR OR attempt to find where (M1)
hour A1
[2 marks]
recognizing (M1)
(A1)
volume eaten is 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.
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.