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Answer :
Let's work through the problem step by step.
### a) Find the [tex]\( I \)[/tex]-intercept and interpret its meaning.
The [tex]\( I \)[/tex]-intercept is the value of the total annual net income [tex]\( I \)[/tex] when [tex]\( t = 0 \)[/tex]. To find the [tex]\( I \)[/tex]-intercept, substitute [tex]\( t = 0 \)[/tex] into the function [tex]\( I = f(t) = 90.3 \sqrt{t} - 98.2 \)[/tex].
Since [tex]\( \sqrt{0} = 0 \)[/tex], we have:
[tex]\[ I = 90.3 \times 0 - 98.2 = -98.2. \][/tex]
Interpretation: The [tex]\( I \)[/tex]-intercept is -98.2 billion dollars. This implies that according to the model, the projected net income was -98.2 billion dollars in the year 2000. However, a negative net income is unrealistic, suggesting that the model may not be meaningful for very early years close to 2000.
### b) Find [tex]\( I(10) \)[/tex] and interpret its meaning.
To find [tex]\( I(10) \)[/tex], substitute [tex]\( t = 10 \)[/tex] into the function:
[tex]\[ I(10) = 90.3 \times \sqrt{10} - 98.2. \][/tex]
This calculates to approximately 187.35 billion dollars.
Interpretation: In the year 2010 (10 years after 2000), the model predicts that the total annual net income of the top five oil companies was about 187.35 billion dollars.
### c) Use the model to estimate when the total annual net income was $144,000,000,000.
To find when the income was 144 billion dollars, set [tex]\( f(t) = 144 \)[/tex] and solve for [tex]\( t \)[/tex]:
[tex]\[ 90.3 \sqrt{t} - 98.2 = 144. \][/tex]
Solving this equation gives [tex]\( t \approx 7.19 \)[/tex].
Interpretation: The model estimates that the total annual net income was 144 billion dollars approximately 7.19 years after the year 2000, which would be around the year 2007.
These are the solutions and their interpretations for each part of the problem.
### a) Find the [tex]\( I \)[/tex]-intercept and interpret its meaning.
The [tex]\( I \)[/tex]-intercept is the value of the total annual net income [tex]\( I \)[/tex] when [tex]\( t = 0 \)[/tex]. To find the [tex]\( I \)[/tex]-intercept, substitute [tex]\( t = 0 \)[/tex] into the function [tex]\( I = f(t) = 90.3 \sqrt{t} - 98.2 \)[/tex].
Since [tex]\( \sqrt{0} = 0 \)[/tex], we have:
[tex]\[ I = 90.3 \times 0 - 98.2 = -98.2. \][/tex]
Interpretation: The [tex]\( I \)[/tex]-intercept is -98.2 billion dollars. This implies that according to the model, the projected net income was -98.2 billion dollars in the year 2000. However, a negative net income is unrealistic, suggesting that the model may not be meaningful for very early years close to 2000.
### b) Find [tex]\( I(10) \)[/tex] and interpret its meaning.
To find [tex]\( I(10) \)[/tex], substitute [tex]\( t = 10 \)[/tex] into the function:
[tex]\[ I(10) = 90.3 \times \sqrt{10} - 98.2. \][/tex]
This calculates to approximately 187.35 billion dollars.
Interpretation: In the year 2010 (10 years after 2000), the model predicts that the total annual net income of the top five oil companies was about 187.35 billion dollars.
### c) Use the model to estimate when the total annual net income was $144,000,000,000.
To find when the income was 144 billion dollars, set [tex]\( f(t) = 144 \)[/tex] and solve for [tex]\( t \)[/tex]:
[tex]\[ 90.3 \sqrt{t} - 98.2 = 144. \][/tex]
Solving this equation gives [tex]\( t \approx 7.19 \)[/tex].
Interpretation: The model estimates that the total annual net income was 144 billion dollars approximately 7.19 years after the year 2000, which would be around the year 2007.
These are the solutions and their interpretations for each part of the problem.
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