Chapter 7.1, Problem 4P

To determine

To find: The values for number of insects for each $-2$ , $2$ and $0$ number of weeks and their representation in the particular situations.

Answer to Problem 4P

The number of insects for each $-2$ , $2$ and $0$ number of weeks are $600$ , $48600$ and $5400$ respectively.

Explanation of Solution

Given information:

The expression for number of insects is $5400\cdot 3^w$ , the number of weeks after the population measured be $w$ and the population of insects triples per week.

Calculation:

Let number of insects be $n$ .

The given expression for number of insects is:

  $n=5400\cdot 3^w$

To determine number of insects for $-2$ weeks, substitute $-2$ for $w$ in the above expression.

  $n=5400\cdot 3^{-2}$

Formula for exponents raised to the negative power equals is:

  $a^{-n}=\frac{1}{a^n}$

To calculate the value of $3^{-2}$ , substitute $2$ for $n$ and $3$ for $a$ in the above formula.

  $\begin{array}{c}{3}^{-2}=\frac{1}{3^2}\\ =\frac{1}{3\times 3}\\ =\frac{1}{9}\end{array}$

Substitute $\frac{1}{9}$ for $3^{-2}$ in the above number of insects expression.

  $\begin{array}{c}n=5400\cdot \frac{1}{9}\\ =\frac{5400}{9}\\ =600\end{array}$

So, the number of insects for $-2$ weeks is $600$ which implies that the population of insects reduces nine times for $-2$ weeks instead of increase of nine times.

To determine number of insects for $0$ weeks, substitute $0$ for $w$ in the above given expression of number of insects.

  $n=5400\cdot 3^0$

The property of zero exponents says that any non-zero expression raised to the power of $0$ always equals to $1$ .

Apply the above zero exponent property in the above expression.

  $\begin{array}{c}n=5400\cdot 1\\ =5400\end{array}$

So, the number of insects for $0$ weeks is $5400$ which implies that the population of insects remains same for $0$ weeks. The population triples every week but there is population counted for no weeks so it remains same.

To determine number of insects for $2$ weeks, substitute $2$ for $w$ in the above given expression of number of insects.

  $\begin{array}{c}n=5400\cdot 3^2\\ =5400\cdot 3\cdot 3\\ =48600\end{array}$

So, the number of insects for $2$ weeks is $48600$ which implies that the population of insects increases nine times for $2$ weeks. The population triples every week, thus for two weeks population of insects increases nine times.

Therefore, the number of insects for each $-2$ , $2$ and $0$ number of weeks are $600$ , $48600$ and $5400$ respectively.