Higher November 2020 Unit 1 Q10
10. A farmer knows that the time, \(t\), taken by goats to eat all the grass in a particular field is inversely proportional to the number of goats, \(g\), in the field.
When there are 25 goats in the field, the time taken to eat all the grass is 36 days.
You may assume that all the goats eat grass at the same rate.
(a) Find a formula for the time, \(t\), in terms of the number of goats, \(g\). [3]
(b) Hence, find the time taken for all of the grass to be eaten when there are 20 goats in the field. [1]
(c) The farmer needs the grass to last for at least 40 days.
What is the greatest number of goats that should be allowed in the field? [2]
What is the greatest number of goats that should be allowed in the field? [2]
| Scheme | Marks | Comments |
|---|---|---|
| \(t\ \alpha\ 1/g\) OR \(t = k/g\) | B1 | Allow \(t\ \alpha\ k/g\) |
| \(36 = k/25\) OR \(k = 900\) | M1 | FT from \(y\ \alpha\ 1/x^n\) with \(n \ne 1\), \(n > 0\) No FT from direct proportion M1 implies B1. |
| \(t = 900/g\) | A1 | May be seen explicitly in part (b). Do not allow \(t\ \alpha\ 900/g\) for the A mark |
| Scheme | Marks | Comments |
|---|---|---|
| (900/20 =) 45 (days) | B1 | FT ‘their formula’ only if non-linear. |
| Scheme | Marks | Comments |
|---|---|---|
| Sight of 900/40 | M1 | FT ‘their formula’ only if non-linear and of equivalent difficulty |
| 22 (goats) | A1 | M1 A0 for an answer of 22·5 or 23 For A1, FT for equivalent difficulty i.e. need to round down an answer with a decimal part of 0.5 or over. Allow use of trial and improvement for M1, provided 22 or 23 seen. A0 for incorrect working e.g. 90/4 given as 22.2, leading to 22. |