1. a.
b. A = 3.2 and n = 1.4
giving y = 3.2x1.4
c. When x = 9
y = 3.2 x 91.4 = 69.36
2. a. y = axb ⇒ log y = log axb = log a + b log x
(This is of the form c + m log x so log y is a linear function of log x)
b. (i)
|
log x
|
log y
|
|
| Human |
4.20
|
7.40
|
| Pony |
5.60
|
8.39
|
| Cow |
6.21
|
8.88
|
(ii) The straight line through the points confirms that log y is a linear function of log x and hence the relation y = axb is justified using the result from part a.
(iii) a is approximately 66.7 and b is approximately 0.75
(iv) y is approximately 1181
(v) % error is 10.9% (Note that the formal treatement of error is no longer in the syllabus.)
(vi) x = 0.535 kg
3. a.
|
Time since birth (x months)
|
Average head circumference (y cm)
|
loge x (to 2DP)
|
logey (to 3 DP)
|
|
1
|
36.6
|
0.00
|
3.600
|
|
2
|
39.3
|
0.69
|
3.671
|
|
3
|
41.0
|
1.10
|
3.714
|
|
4
|
42.1
|
1.39
|
3.740
|
|
6
|
43.7
|
1.79
|
3.777
|
|
8
|
45.0
|
2.08
|
3.807
|
|
12
|
46.9
|
2.48
|
3.848
|
b. Graph is a straight line
c. Graph is a straight line i.e log y = m log x + c
d. a = 36.6 and p = 0.1
e. It is not accurate since it predicts that the circumference of the baby's head at birth is 0 cm, which is impossible.
f. Rate = 0.6697 cm / month
g. The formulae have the same value at x = 18.77 months.
