Chapter 8, Problem 16PT

(a)

To determine

To find: The situation for the number is declining.

Answer to Problem 16PT

The numbers of forms are declining with time.

Explanation of Solution

Given:

The equation is given as,

  $y=3962520{\left(0.98\right)}^x$

Calculation:

The coefficient of the x is 0.98 which means that as the value of x increases, it decreases due to which y also decreases. The given expression states that the numbers of forms are declining with time.

Thus, the numbers of forms are declining with time.

(b)

To determine

To find: The rate at which number is declining

Answer to Problem 16PT

The rate at which number is declining is 1.99%

Explanation of Solution

Given:

The equation is given as,

  $y=3962520{\left(0.98\right)}^x$

Calculation:

Calculate the number of farms in 1960.

  $\begin{array}{c}y=3962520{\left(0.98\right)}^0\\ =3962520\left(1\right)\\ =3962520\end{array}$

Calculate the number of farms in 1961.

  $\begin{array}{c}y=3962520{\left(0.98\right)}^1\\ =3962520\left(0.98\right)\\ =3883270\end{array}$

Calculate the rate.

  $\begin{array}{c}r=\left(\frac{3962520-3883270}{3962520}\right)\left(100\right)\\ =1.99\%\end{array}$

Thus, the rate at which number is declining is 1.99%

(c)

To determine

To find: The prediction when the number of farms less than 1 million.

Answer to Problem 16PT

The population will be less than 1 million before 68 years.

Explanation of Solution

Given:

The equation is given as,

  $y=3962520{\left(0.98\right)}^x$

Calculation:

According to the question,

  $\begin{array}{c}1000000<3962520{\left(0.98\right)}^t\\ \frac{1000000}{3962520}<\frac{3962520}{3962520}{\left(0.98\right)}^t\\ 0.252<{\left(0.98\right)}^t\\ \ln 0.252<t\ln 0.98\end{array}$

Solve further as,

  $\begin{array}{c}t<\left(\frac{1.38}{0.0202}\right)\\ <68.32\\ <68\end{array}$

Thus, the population will be less than 1 million before 68 years.