Chapter 8.3, Problem 57PPS

(a)

To determine

To Find: The values of $S(3),S(15),S(63)$

Answer to Problem 57PPS

  $S(3)=30$ , $S(15)=50$ , $S(63)=70$

Explanation of Solution

Given: $S(a)=10+20{\log }_4(a+1)$

Given: $S(a)=10+20{\log }_4(a+1)$

When $a=3$

  $\begin{array}{l}S(a)=10+20{\log }_4(a+1)\\ S(3)=10+20{\log }_4(3+1)\\ S(3)=10+20{\log }_{4}4\\ S(3)=10+20\\ S(3)=30\end{array}$

When $a=15$

  $\begin{array}{l}S(a)=10+20{\log }_4(a+1)\\ S(15)=10+20{\log }_4(15+1)\\ S(15)=10+20{\log }_{4}16\\ S(15)=10+40\\ S(15)=50\end{array}$

When $a=63$

  $\begin{array}{l}S(a)=10+20{\log }_4(a+1)\\ S(63)=10+20{\log }_4(63+1)\\ S(63)=10+20{\log }_{4}64\\ S(63)=10+20{\log }_4{4}^3\\ S(63)=10+60\\ S(63)=70\end{array}$

Hence, $S(3)=30$ , $S(15)=50$ , $S(63)=70$

(b)

To determine

To Interpret: The meaning of the function.

Answer to Problem 57PPS

  $\$3$ is spent on advertising, $\$30000$ returned in sales.

  $\$15$ is spent on advertising, $\$50000$ returned in sales.

  $\$63$ is spent on advertising, $\$70000$ returned in sales.

Explanation of Solution

Given: $S(a)=10+20{\log }_4(a+1)$

Given: $S(a)=10+20{\log }_4(a+1)$

  $S(3)=30$ : $\$3$ is spent on advertising, $\$30000$ returned in sales.

  $S(15)=50$ : $\$15$ is spent on advertising, $\$50000$ returned in sales.

  $S(63)=70$ : $\$63$ is spent on advertising, $\$70000$ returned in sales

(c)

To determine

To Draw: The graph of the function.

Answer to Problem 57PPS

  

Explanation of Solution

Given: $S(a)=10+20{\log }_4(a+1)$

Given: $S(a)=10+20{\log }_4(a+1)$

The graph of the function is

  

(d)

To determine

To Explain: Why the money spent in advertisement is less efficient as it is used in large amounts.

Answer to Problem 57PPS

Fixed increase in dollar advertisement increases sales by diminishing amount as advertisement increases.

Explanation of Solution

Given: $S(a)=10+20{\log }_4(a+1)$

Given: $S(a)=10+20{\log }_4(a+1)$

The graph of the function is

  

From the graph it is seen that as the fixed increase in dollar advertisement increases sales by diminishing amount as advertisement increases.