Chapter 7, Problem 8QC

(a)

To determine

To Simplify :

The division $\frac{4000x^2}{x^2+50}$

Answer to Problem 8QC

  $4000-\frac{200,000}{x^2+50}$

Explanation of Solution

Given Information:

The number of specialty coffee mugs sold at a coffee shop can be estimated by $n=\frac{4000x^2}{x^2+50}$ , where x is the amount of money spent on advertising in hundreds of dollars and n is the number of mugs sold.

Calculation:

  $\begin{array}{l}Given\\ \frac{4000x^2}{x^2+50}\\ =\frac{4000(x^2+50-50)}{x^2+50}\\ =\frac{4000(x^2+50)-4000(50)}{x^2+50}\\ =\frac{4000(x^2+50)}{x^2+50}-\frac{200,000}{x^2+50}\\ =4000-\frac{200,000}{x^2+50}\\ Hence,\\ \frac{4000x^2}{x^2+50}=4000-\frac{200,000}{x^2+50}\end{array}$

(b)

To determine

To Find :

The number of mugs sold if $1000 is spent on advertising.

Answer to Problem 8QC

  $n\approx 2667$

Explanation of Solution

Given Information:

The number of specialty coffee mugs sold at a coffee shop can be estimated by $n=\frac{4000x^2}{x^2+50}$ , where x is the amount of money spent on advertising in hundreds of dollars and n is the number of mugs sold. $1000 is spent on advertising.

Calculation:

From part (a), we have

  $\frac{4000x^2}{x^2+50}=4000-\frac{200,000}{x^2+50}$

  $\begin{array}{l}\text{Given }\\ \text{Amount spent on advertising = \$1000}\\ \text{x is the amount spent in hundreds of dollars on advertising}\\ \text{So, x=10}\\ \text{Substituting in the equation and solving for n:}\\ n=\frac{4000x^2}{x^2+50}=4000-\frac{200,000}{x^2+50}\\ \Rightarrow n=4000-\frac{200,000}{{(10)}^2+50}\\ \Rightarrow n=4000-\frac{200,000}{100+50}\\ \Rightarrow n\approx 4000-1333.334\\ \Rightarrow n\approx 2667\\ So,\text{ the number of mugs sold is about 2667}\text{.}\end{array}$