Chapter 6.6, Problem 51E

(a)

To determine

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

Answer to Problem 51E

The next term will be multiplied by $\frac{3}{4}$ and so on:

1536, 1152, 864, 648, 486

Explanation of Solution

Given information:

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

Formula used:

1 gallon = 128 fluid ounces

Calculation:

1 gallon = 128 fluid ounces

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

  $a_1=16\left(\frac{3}{4}\right)=12gallons=1536fl.oz$

Therefore, next term will be multiplied by $\frac{3}{4}$ and so on:

1536, 1152, 864, 648, 486

Conclusion:

The next term will be multiplied by $\frac{3}{4}$ and so on:

1536, 1152, 864, 648, 486

(b)

To determine

To find: The nth term of the sequence.

Answer to Problem 51E

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

Explanation of Solution

Given information:

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

Formula used:

  $a_n=a_1{r}^{n-1}$

Calculation:

Use the formula

  $a_n=a_1{r}^{n-1}$

From $\left(a\right),a_1=1536\&r=\frac{3}{4}$ so:

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

Conclusion:

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

(c)

To determine

To find: The time when the soup will run out.

Answer to Problem 51E

There is always three quarters left

Explanation of Solution

Given information:

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

Formula used:

  $a_n=a_1{r}^{n-1}$

Calculation:

Given that there is always three quarters left, then mathematically, there will always be a soup.

Conclusion:

There is always three quarters left