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Methane is a greenhouse gas implicated as a contributor to global warming. Given below is the table of values of [tex]Q = w(t)[/tex], the atmospheric methane level in parts per billion (ppb) [tex]t[/tex] years after 1980.

[tex]\[
\begin{array}{c|c|c|c|c|c|c|c}
\hline
t & 0 & 5 & 10 & 15 & 20 & 25 & 30 \\
\hline
Q & 1575 & 1660 & 1715 & 1750 & 1770 & 1775 & 1785 \\
\hline
\end{array}
\][/tex]

(a) Evaluate [tex]\frac{w(5) - w(0)}{5 - 0}[/tex] and [tex]\frac{w(15) - w(5)}{15 - 5}[/tex]. Round your answers to one decimal place.

[tex]\[
\begin{array}{l}
\frac{w(5) - w(0)}{5 - 0} = \frac{1660 - 1575}{5} = 17.0 \\
\frac{w(15) - w(5)}{15 - 5} = \frac{1750 - 1660}{10} = 9.0
\end{array}
\][/tex]

(b) Compare your answers in (a). What does this tell you about atmospheric methane levels? Enter a year in each box to complete the explanation.

This tells us that the methane level rises at a slower average rate from the years [tex] \square [/tex] to [tex] \square [/tex] than from the years [tex] \square [/tex] to [tex] \square [/tex].

[tex]\[
\begin{array}{l}
\text{This tells us that the methane level rises at a slower average rate from the years } \boxed{5} \text{ to } \boxed{15} \text{ than from the years } \boxed{0} \text{ to } \boxed{5}.
\end{array}
\][/tex]

Sagot :

GinnyAnswer
Alright, let's go through the problem step-by-step:

#### (a) Evaluate [tex]\(\frac{w(5) - w(0)}{5 - 0}\)[/tex] and [tex]\(\frac{w(15) - w(5)}{15 - 5}\)[/tex]

We have the table of values:

[tex]\[ \begin{array}{c|c|c|c|c|c|c|c} \hline t & 0 & 5 & 10 & 15 & 20 & 25 & 30 \\ \hline Q & 1575 & 1660 & 1715 & 1750 & 1770 & 1775 & 1785 \\ \hline \end{array} \][/tex]

1. First, we calculate the average rate of change between [tex]\( t = 0 \)[/tex] and [tex]\( t = 5 \)[/tex]:

[tex]\[ \frac{w(5) - w(0)}{5 - 0} = \frac{1660 - 1575}{5 - 0} \][/tex]

[tex]\[ = \frac{85}{5} = 17.0 \][/tex]

2. Next, calculate the average rate of change between [tex]\( t = 5 \)[/tex] and [tex]\( t = 15 \)[/tex]:

[tex]\[ \frac{w(15) - w(5)}{15 - 5} = \frac{1750 - 1660}{15 - 5} \][/tex]

[tex]\[ = \frac{90}{10} = 9.0 \][/tex]

So, the average rates of change are:

[tex]\[ \frac{w(5) - w(0)}{5 - 0} = 17.0 \quad \text{ppb per year} \][/tex]

[tex]\[ \frac{w(15) - w(5)}{15 - 5} = 9.0 \quad \text{ppb per year} \][/tex]

#### (b) Compare your answers in (a). What does this tell you about atmospheric methane levels?

We notice that the rate of change [tex]\(\frac{w(5) - w(0)}{5 - 0}\)[/tex] is higher than the rate of change [tex]\(\frac{w(15) - w(5)}{15 - 5}\)[/tex].

This tells us that the methane level rises at a slower average rate from the years 5 to 15 than from the years 0 to 5.

To complete the explanation:

[tex]\[ \text{This tells us that the methane level rises at a slower average rate from the years} \, 0 \, \text{to} \, 5 \, \text{than from the years} \, 5 \, \text{to} \, 15. \][/tex]

So, the final answer is:

[tex]\[ \text{This tells us that the methane level rises at a slower average rate from the years} \, 0 \, \text{to} \, 5 \, \text{than from the years} \, 5 \, \text{to} \, 15. \][/tex]
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