Chapter 7.1, Problem 46PPE

To determine

To Evaluate: The expression $45\cdot {1.5}^d$ for $d=-2,d=0,d=1$ and to write what does each vale of the expression represents.

Answer to Problem 46PPE

The required answer is 20,45 and 67.5.

Explanation of Solution

Given information:

A Galapagos cactus finch population increases by half every decade. The number of finches is modeled by the expression $45\cdot {1.5}^d$ , where d is the number of decades after the population was measured.

Calculation:

  $\begin{array}{l}45\cdot {1.5}^d\\ =45\cdot {1.5}^{-2}\end{array}$

(Substitute -2 for d)

  $\begin{array}{l}=\frac{45}{{1.5}^2}\\ =20\end{array}$

(Simplify)

The value of the expression for $d=-2$ is 20.

There were 20 finches 2 decades before the population was measured.

  $\begin{array}{l}45\cdot {1.5}^d\\ =45\cdot {1.5}^0\end{array}$

(Substitute 0 for d)

  $\begin{array}{l}=45\cdot 1\\ =45\end{array}$

(Simplify)

The value of the expression for $d=0$ is 45.

There were 45 finches at the time when population was measured.

  $\begin{array}{l}45\cdot {1.5}^d\\ =45\cdot {1.5}^1\end{array}$

(Substitute 1 for d)

  $\begin{array}{l}=45\cdot 1.5\\ =67.5\end{array}$

(Simplify)

The value of the expression for $d=1$ is 67.5.

There were 67.5 finches 1 decade after the population was measured.