Chapter 9.3, Problem 48PPSE

Solution

a.

To write a sequence of number to represent the situation.

  $5000,3750,2812.5\dots$

Given:

The balloon filled on the first day is $5000{\text{ cm}}^{\text{3}}$ .

Calculation:

The volume of the balloon on the first day is:

  $a_1=5000$

On the second day, the volume of the balloon is:

  $\begin{array}{l}{a}_2=5000\times \frac{3}{4}\\ \Rightarrow a_2=3750\end{array}$

On the third day, the volume of the balloon is:

  $\begin{array}{l}{a}_3=3750\times \frac{3}{4}\\ \Rightarrow a_3=2812.5\end{array}$

The sequence is:

  $5000,3750,2812.5\dots$

b.

To find the type of the given sequence.

Geometric Sequence

In the given sequence,the ratio of the successive terms is constant and the common ratio $\left(r\right)$ is $\frac{3}{4}.$ So, the given sequence is a geometric sequence.

c.

To find the left amount of helium at the start of the tenth day.

The helium left at the start of the tenth day is $375.4{\text{ cm}}^{\text{3}}$ .

Given:

The sequence is:

  $5000,3750,2812.5,\dots \dots$

Calculation:

The explicit formula for the geometric sequence is:

  $a_n=a_1{r}^{n-1}$

Now, calculating the left amount of helium at the start of tenth day as:

  $\begin{array}{l}{a}_n=a_1{r}^{n-1}\\ \Rightarrow a_{10}=5000{\left(\frac{3}{4}\right)}^{10-1}\\ \Rightarrow a_{10}=5000{\left(\frac{3}{4}\right)}^9\\ \Rightarrow a_{10}=5000\times \left(0.07508\right)\\ \Rightarrow a_{10}=375.4\end{array}$

Hence, the helium left in the balloon at the start of the tenth day is $375.4{\text{ cm}}^{\text{3}}$ .