Chapter 6.3, Problem 13Q

Solution

(a)

To plot the points given in table

Given:

The table showing the amount y (in grams) of an element remaining in a jar after x centuries

Graph:

Graph of points plotted is as follows:

(b)

To tell whether the data can be modelled by a linear or an exponential function, then write the function

Data can be modelled by exponential function

Required exponential function is $y=50e^{-0.6931x}$

Given:

The table showing the amount y (in grams) of an element remaining in a jar after x centuries

Concept Used:

A linear function is a function whose graph is a straight line

An exponential function has a non linear graph.

It is of the form $y=A{e}^{kx}$ where A and k are constants

Calculation:

From graph it is clear that it is a non linear graph.

Hence,

Given function must be an exponential function

First data gives $\left(x,y\right)\to \left(0,50\right)$

This means that if the function is exponential then it must be $50e^{kx}$

(Because then only $y=50$ )

Now for same data $\left(1,25\right)$

  $\begin{array}{l}50e^{k\times 1}=25\\ e^k=\frac{25}{50}\\ e^k=\frac{1}{2}\\ \text{taking }\ln \text{ both sides}\\ \text{ln}\left(e^k\right)=\ln \left(\frac{1}{2}\right)\\ k=-0.6931\end{array}$

Conclusion:

Therefore,

Required exponential function is $y=50e^{-0.6931x}$

(c)

To find how much element remains after 600 years

Quantity (in grams) of remaining element after 600 years = ( $1.239558e-179$ ) grams

Given:

x=600 years

Concept Used:

Put x=600 in $y=50e^{-0.6931x}$

Calculation:

  $\begin{array}{l}y=50\times e^{-0.6931\times 600}\\ y=50e^{-415.86}\\ y=1.239558e-179\end{array}$

Conclusion:

Quantity (in grams) of remaining element after 600 years = ( $1.239558e-179$ ) grams