Chapter 6.5, Problem 51E

a

To determine

To Write the first five terms of the sequence of the number of fluid ounces of soup left each day.

Answer to Problem 51E

The first five terms of the sequence of the number of fluid ounces of soup left each day are $1536,1152,\text{ 864},\text{ 648},\text{ 486 }$

Explanation of Solution

Given information:

A soup kitchen makes 16 gallons of soup. Each day, a quarter of the soup is served and the rest is saved and the rest is saved for the next day.

  

  $\text{1}$ gallon = $\text{128}$ fluid ounces

On the first day, a quarter is served so three quarters are left:

  ${\text{a}}_1=16\left(\frac{3}{4}\right)$

  = $12$ gallons

  = $1536$ fluid ounces

Therefore,

Next term will be multiplied by $\frac{3}{4}$ and so on: $1536,1152,\text{ 864},\text{ 648},\text{ 486 }$

b

To determine

To Write an equation that represents the $n$ th term of the sequence.

Answer to Problem 51E

The $n$ th term of the sequence is:

  ${\text{a}}_{\text{n}}=1536{\left(\frac{3}{4}\right)}^{n-1}$

Explanation of Solution

Given information:

A soup kitchen makes 16 gallons of soup. Each day, a quarter of the soup is served and the rest is saved and the rest is saved for the next day.

  

  $\text{1}$ gallon = $\text{128}$ fluid ounces

The given sequence is following geometric series.

For $n$ th term, ${\text{a}}_{\text{n}}={\text{ a}}_1{\text{r}}^{n-1}$

From subpart (a), ${\text{a}}_1=1536$ and $r=\frac{3}{4}$

Therefore,

  ${\text{a}}_{\text{n}}=1536{\left(\frac{3}{4}\right)}^{n-1}$

c

To determine

To explain, When is all the soup gone?

Answer to Problem 51E

It is stated in the question that there is always three quarters left, then mathematically, there will always be a soup

Explanation of Solution

Given information:

A soup kitchen makes 16 gallons of soup. Each day, a quarter of the soup is served and the rest is saved and the rest is saved for the next day.

  

  $\text{1}$ gallon = $\text{128}$ fluid ounces

It is stated in the question that there is always three quarters left, then mathematically, there will always be a soup