Chapter 4, Problem 21TP

Solution

To Graph: a scatter plot of given points and also find the exponential function of the given point.

  $y=5996.912\cdot {\left(1.3893\right)}^x$ .

Given: points are given $\left(x,y\right)$ as $\left(1974,6000\right)$ , $\left(1979,29000\right)$ , $\left(1982,134000\right)$ , $\left(1985,275000\right)$ , $\left(1989,1200000\right)$ , $\left(1993,3100000\right)$

  $\left(1997,7500000\right)$ , $\left(1999,9500000\right)$ , $\left(2000,42000000\right)$ and $\left(2004,125000000\right)$ .

Concept used:

(1) If a function $y=f\left(x\right)$ passes through a point $\left(x_0,y_0\right)$ , then the point $\left(x_0,y_0\right)$ satisfies the equation of the function, that is, $y_0=f\left(x_0\right)$ .

(2) An exponential model or function is as follows:

  $y=a\cdot {(b)}^x$ ; where $a\ne 0$ .

(3) Equation of a line passing through the given two points as like $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ .

Then, $\left(y-y_1\right)=m\left(x-x_1\right)$ .

Here, $m$ is the slope. And $m$ is as follows:

  $m=\frac{y_2-y_1}{x_2-x_1}$ .

Calculation:

(a)The table of data pairs $\left(x,\ln y\right)$ written as;

  $\begin{array}{ccccccccccc}x& 0& 5& 8& 11& 15& 19& 23& 25& 26& 30\\ \ln y& 8.699& 10.275& 11.806& 12.524& 13.997& 14.947& 15.831& 16.067& 17.533& 18.644\end{array}$

The Scatter graph of $\left(x,\ln y\right)$ is given as;

  

Here, choose two points $\left(0,8.699\right)$ and $\left(19,14.947\right)$ then the equation of the line;

  $\ln y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$

Now, substitute the value;

  $\begin{array}{c}\ln y-8.699=\frac{14.947-8.699}{19-0}\left(x-0\right)\\ \ln y-8.699=\frac{6.248}{19}\cdot x\\ \ln y-8.699=0.3288\cdot x\end{array}$

Thus,

  $\ln y=0.3288\cdot x+8.699$

  ....... (1)

(B)

Now, take the exponentiation of each side by e of the above equation (1);

  $\begin{array}{c}{e}^{\ln \left(y\right)}=e^{0.3288x+8.699}\left[e^{\ln a}=a\right]\\ y=e^{0.3288x}\cdot e^{8.699}\\ y\approx {\left[{\left(e\right)}^{0.3288}\right]}^x\cdot 5996.912\left[\begin{array}{c}e\approx 2.7182\\ e^{0.3288}\approx 1.3893\\ e^{8.699}\approx 5996.912\end{array}\right]\\ y\approx 5996.912\cdot {\left(1.3893\right)}^x\end{array}$

Therefore, the exponential function is the model of the data pairs $y=5996.912\cdot {\left(1.3893\right)}^x$ .

(C) From the exponential model is defined as follows:

  $y=5996.912\cdot {\left(1.3893\right)}^x$   ....... (2)

Now find the total number of years, so;

  $2008-1974=34\text{ years}$ .

Here, the $x$ be the represents the years.

Therefore, $x=34$ substitute in the above exponential model equation $y=e^{0.3288x+8.699}$ .

Then,

  $\begin{array}{c}y=e^{0.3288\cdot \left(34\right)+8.699}\\ y=e^{11.1792+8.699}\\ y=e^{19.8782}\end{array}$

Thus,

  $y\approx 428644706.30558$

At our convenience, neglect the term after the decimal.

  $y\approx 428644706$

This is a predicted number of components in $2008$ is approximate.

(D) Moore’s law:- The law states that the number of components would double every $18$ month using the model obtained, we can calculate an example for the statement. Where we could have a first year and a year after $18$ months.

  $y=e^{0.3288x+8.699}$

Now, calculate the first year put $x=1$ .

  $\begin{array}{c}y=e^{0.3288\cdot (1)+8.699}\\ y=e^{0.3288+8.699}\\ y=e^{9.0278}\\ y\approx 8323.7151\end{array}$

At our convenience, neglect the term after the decimal.

  $y=8323$ components.

Then, calculate the year after $18$ month. So, $x=2.5$ substitute in the exponential model equation $y=e^{0.3288x+8.699}$ .

  $\begin{array}{c}y=e^{0.3288\cdot (2.5)+8.699}\\ y=e^{0.822+8.699}\\ y=e^{9.521}\\ y\approx 13629.786\end{array}$

At our convenience, neglect the term after the decimal.

  $y=13629$ components.

Therefore, divide the year after $18$ months from the first year to see if the number of components defines as follows:

  $\frac{13629}{8323}\approx 1.64$

According to the model, Moore’s law does not exist or hold. Because the number of components did not double.

Conclusion:

Hence, the exponential function of the model satisfies the given data pairs as follows: $y=5996.912\cdot {\left(1.3893\right)}^x$ and also represents the data pairs in the scatter plot. Gordon Moore's law does not hold or exist. Because the number of components(transistors) did not double.