Chapter 8.6, Problem 39PPS

(a).

To determine

To find: The how many pet owners were there in 2010.

Answer to Problem 39PPS

The number of pet owners in the year $2010$ is approximately $62.737$ thousand

Explanation of Solution

Given information:

The numbers of pet owners $n$ in thousands after $t$ years

  $n=35\left[{\log }_4\left(t+2\right)\right]$

  $t=0$ represents the year $2000$ .

Calculation:

Consider the numbers of pet owners $n$ in thousands after $t$ years are modeled by:

  $n=35\left[{\log }_4\left(t+2\right)\right]$

Where, $t=0$ represents the year $2000$ , consider $n_1$ , is the number of pet owners in the year $2010$ that is $n=n_1$ for $t=10$ therefore:

  $\begin{array}{l}n=35\left[{\log }_4\left(t+2\right)\right]\\ =35\left[{\log }_{4}12\right]\\ =35\frac{{\log }_{10}12}{{\log }_{10}4}\\ \approx 62.737\end{array}$

Use TI-83 calculator to solve the screenshot is given as below:

  

Therefore, the number of pet owners in the year $2010$ is approximately $62.737$ thousand.

(b).

To determine

To find: The how long there are 80,000 pet owners.

Answer to Problem 39PPS

In $2022$ there are 80,000 pet owners.

Explanation of Solution

Given information:

The numbers of pet owners $n$ in thousands after $t$ years

  $n=35\left[{\log }_4\left(t+2\right)\right]$

  $t=0$ Represents the year $2000$ .

Calculation:

Consider the numbers of pet owners $n$ in thousands after $t$ years are modeled by:

  $n=35\left[{\log }_4\left(t+2\right)\right]$

Where, $t=0$ represents the year $2000$ , consider $n_1$ , is the number of pet owners in the year $2010$ that is $n=n_1$ for $t=10$ therefore:

  $\begin{array}{c}n=35\left[{\log }_4\left(t+2\right)\right]\\ n=35\left[{\log }_4(t+2)\right]\\ \frac{80}{35}=\left[{\log }_4(t+2)\right]\end{array}$

  $\begin{array}{c}\frac{80}{35}=\frac{\log (t+2)}{\log 4}\\ t+2={10}^{\frac{80}{35}\log 4}\\ t=22\end{array}$

Therefore, in $2022$ there are 80,000 pet owners.