Chapter 3.8, Problem 32E

(a)

To determine

To find: The sensitivity of the body.

Answer to Problem 32E

Sensitivity of the body is $S=\frac{-54.4x^{-0.6}}{\left(1+16x^{0.8}+8x^{0.4}\right)}$.

The graph is plotted between x and y coordinates for both $R$ and $S$ is shown in figure (1).

Explanation of Solution

Given:

The experimental formula is shown below.

$R=\frac{40+24x^{0.4}}{1+4x^{0.4}}$ (1)

Calculation:

Calculate the sensitivity $S$.

$S=\frac{dR}{dx}$

Sensitivity is defined to be the rate of change of the reaction with respect to $x$.

Substitute $\frac{40+24x^{0.4}}{1+4x^{0.4}}$ for $R$ in the above equation.

$S=\frac{d}{dx}\left(\frac{40+24x^{0.4}}{1+4x^{0.4}}\right)$

Apply the quotient rule below.

$\left(\frac{u}{v}\right)\text{'}=\frac{u\text{'}v-v\text{'}u}{v^2}$

Substitute $\left(40+24x^{0.4}\right)$ for $u$ and $\left(1+4x^{0.4}\right)$ for $v$ in the above equation.

$\begin{array}{c}S=\frac{\left(40+24x^{0.4}\right)\text{'}\left(1+4x^{0.4}\right)-\left(1+4x^{0.4}\right)\text{'}\left(40+24x^{0.4}\right)}{{\left(1+4x^{0.4}\right)}^2}\\ =\frac{9.6x^{-0.6}\left(1+4x^{0.4}\right)-1.6x^{-0.6}\left(40+24x^{0.4}\right)}{{\left(1+4x^{0.4}\right)}^2}\\ =\frac{1.6x^{-0.6}\left(6\left(1+4x^{0.4}\right)-\left(40+24x^{0.4}\right)\right)}{{\left(1+4x^{0.4}\right)}^2}\\ =\frac{1.6x^{-0.6}\left(6+24x^{0.4}-40-24x^{0.4}\right)}{\left(1+{\left(4x^{0.4}\right)}^2+2\left(1\right)\left(4x^{0.4}\right)\right)}\end{array}$

$S=\frac{-54.4x^{-0.6}}{\left(1+16x^{0.8}+8x^{0.4}\right)}$ (2)

Thus, the sensitivity of the body is $\frac{-54.4x^{-0.6}}{\left(1+16x^{0.8}+8x^{0.4}\right)}$.

(b)

To determine

To illustrate: The part (a) by graphing both R and S as functions of x and comment on the values of $R$ and $S$ at low levels of brightness.

Explanation of Solution

Illustration:

Sketch the curve.

Calculate the value of $R$ using the equation (1).

$R=\frac{40+24x^{0.4}}{1+4x^{0.4}}$

Substitute 0 for $x$ in the equation (1).

$\begin{array}{c}R=\frac{40+24{\left(0\right)}^{0.4}}{1+4{\left(0\right)}^{0.4}}\\ =40\end{array}$

Repeat the calculation of the value $R$ for values of $x$ from 0.2 till 10.

Tabulate the value of $x$ and $R$ as shown in table (1).

$x$	$R=\frac{40+24x^{0.4}}{1+4x^{0.4}}$
0.00	40.00
0.20	16.96
0.40	15.01
0.60	13.98
0.80	13.30
1.00	12.80
1.20	12.41
1.40	12.10
1.60	11.83
1.80	11.61
2.00	11.42
2.20	11.24
2.40	11.09
2.60	10.95
2.80	10.83
3.00	10.72
3.20	10.61
3.40	10.52
3.60	10.43
3.80	10.35
4.00	10.27
4.20	10.20
4.40	10.13
4.60	10.06
4.80	10.00
5.00	9.95
5.20	9.89
5.40	9.84
5.60	9.79
5.80	9.74
6.00	9.70
6.20	9.66
6.40	9.62
6.60	9.58
6.80	9.54
7.00	9.50
7.20	9.47
7.40	9.43
7.60	9.40
7.80	9.37
8.00	9.34
8.20	9.31
8.40	9.28
8.60	9.25
8.80	9.22
9.00	9.20
9.20	9.17
9.40	9.15
9.60	9.12
9.80	9.10
10.00	9.08

Calculate the value of $S$ using the equation (2).

$S=\frac{-54.4x^{-0.6}}{\left(1+16x^{0.8}+8x^{0.4}\right)}$

Substitute 0 for $x$ in the equation (1).

$\begin{array}{c}S=\frac{-54.4{\left(0\right)}^{-0.6}}{\left(1+16{\left(0\right)}^{0.8}+8{\left(0\right)}^{0.4}\right)}\\ =0\end{array}$

Repeat the calculation of the value $S$ for values of $x$ from 0.2 till 10.

Tabulate the value of $x$  and $S$ as shown in table (2).

$x$	$S=\frac{-54.4x^{-0.6}}{\left(1+16x^{0.8}+8x^{0.4}\right)}$
0.00	0
0.20	-14.856
0.40	-6.6235
0.60	-4.0713
0.80	-2.8659
1.00	-2.176
1.20	-1.7342
1.40	-1.4297
1.60	-1.2083
1.80	-1.041
2.00	-0.9106
2.20	-0.8064
2.40	-0.7215
2.60	-0.6512
2.80	-0.592
3.00	-0.5417
3.20	-0.4984
3.40	-0.4609
3.60	-0.428
3.80	-0.399
4.00	-0.3733
4.20	-0.3503
4.40	-0.3298
4.60	-0.3112
4.80	-0.2944
5.00	-0.2791
5.20	-0.2651
5.40	-0.2524
5.60	-0.2406
5.80	-0.2298
6.00	-0.2198
6.20	-0.2105
6.40	-0.2019
6.60	-0.1939
6.80	-0.1864
7.00	-0.1795
7.20	-0.1729
7.40	-0.1668
7.60	-0.161
7.80	-0.1556
8.00	-0.1505
8.20	-0.1456
8.40	-0.1411
8.60	-0.1367
8.80	-0.1326
9.00	-0.1288
9.20	-0.1251
9.40	-0.1215
9.60	-0.1182
9.80	-0.115
10.00	-0.112

Graph:

Sketch the curve using table (1) and table (2) as shown in figure (1).

Refer the figure (1).

For all the small values of $x$, we have $R$ reaching a value near 40, which is quite high. So a small stimulus produces a large reaction which is something to except.

Comments:

At low level of brightness, the eye is more sensitive to slight changes than it is at higher level of brightness.