1. The set of values below are to be modelled by an power function y = Axn.
|
x
|
1
|
2
|
3
|
4
|
5
|
|
y
|
3.20
|
8.44
|
14.90
|
22.29
|
30.46
|
a. By plotting ln x against ln y, show that this is an appropriate model.
b. Find the values of n and A.
c. Find the value of y when x = 9
2. a. Show that if y = axb then log y is a linear function of log x.
b. In 1938 Francis Benedict published a comparison of the average heat production of animals of various body weights. Part of his data is given in the following table, which also shows some values of the natural logarithms (loge or ln) of the data.
|
Animal
|
Average weight (x kg) |
Average heat production per day
( y calories) |
log e x (to 2 DP)
|
log e y (to 2 DP)
|
|
Hen
|
2
|
115
|
0.69
|
4.74
|
|
Cat
|
3
|
152
|
1.10
|
5.02
|
|
Dog
|
14
|
485
|
2.64
|
6.18
|
|
Chimpanzee
|
38
|
1090
|
3.64
|
6.99
|
|
Human
|
67
|
1640
|
||
|
Pony
|
270
|
4400
|
||
|
Cow
|
500
|
7200
|
(i) Complete this table by adding the other 6 logarithmic values.
(ii) Draw a log-log graph based on the given data. State why x and y are related (approximately) by a formula of the type y = axb.
(iii) From your graph in part (ii) find values for a and b. Show all of your working.
Note Your values for a and b in part (iii) should be in the region of 68 and 0.75 respectively. For the rest of the question assume that the relation between x and y is exactly y = 68x0.75.
(iv) Estimate the average heat production of sheep with average weight 45 kg.
(v) For very small mammals like mice, rats and ferrets, the relation y = 60x0.55 more accurately fits the relevant data. For a rat with a weight of 0.3 kg, what is the percentage error in using the formula y = 68x0.75 rather than the more precise formula y = 60x0.55? (Note that the formal treatement of error is no longer in the syllabus.)
(vi) At what weight do the two formulae in part (v) give the same average heat production?
3. The average head circumference for a group of babies, measured at certain times during their first 12 months after birth, are given in the table below. the table also shows some values of the natural logarithms (loge) of the data.
|
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
|
||
|
8
|
45.0
|
||
|
12
|
46.9
|
a. Complete the copy of this table by adding the other six logarithm values.
b. Draw the log-log graph based on the given data.
c. From your graph, give clear reasoning as to why x and y are related (approximately) by a formula of the type y = axp. Include a mathematical derivation to back up your argument.
d. From your graph, find values of a and p. Show all working.
e. The model y = axp gives a fairly accurate estimate of the head circumference of a baby between 1 and 12 months old. Why is this model inaccurate for a baby at birth?
f. The head circumference of one baby, Jeremy, was modelled by the formula y = 38x0.09. Using this model, find the rate of growth of the circumference of his head at 6 months.
g. When Jeremy was older, the formula y = 44x0.04 gave a better approximation of his head circumference than the formula in part f. At what age do the two formulae give exactly the same value?
