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The total revenue from the sale of a popular book is approximated by the rational function [tex][tex]$R(x)=\frac{1600 x^2}{x^2+4}$[/tex][/tex], where [tex][tex]$x$[/tex][/tex] is the number of years since publication and [tex][tex]$R(x)$[/tex][/tex] is the total revenue in millions of dollars. Use this function to complete parts (a) through (d).

(a) Find the revenue after 1 year.
(b) Determine the horizontal asymptote of [tex][tex]$R(x)$[/tex][/tex].
(c) Calculate the revenue as [tex][tex]$x$[/tex][/tex] approaches infinity.
(d) Interpret the meaning of the horizontal asymptote in the context of this problem.

Sagot :

GinnyAnswer
Let's solve each part of the given question using the provided rational function [tex]\( R(x) = \frac{1600 x^2}{x^2 + 4} \)[/tex], where [tex]\( x \)[/tex] is the number of years since publication, and [tex]\( R(x) \)[/tex] is the total revenue in millions of dollars.

### Part a: Find [tex]\( R(1) \)[/tex]
To find the revenue after 1 year, substituting [tex]\( x = 1 \)[/tex] into the function [tex]\( R(x) \)[/tex]:

[tex]\[ R(1) = \frac{1600 \cdot 1^2}{1^2 + 4} \][/tex]
[tex]\[ = \frac{1600 \cdot 1}{1 + 4} \][/tex]
[tex]\[ = \frac{1600}{5} \][/tex]
[tex]\[ = 320 \][/tex]

So, the revenue after 1 year is [tex]\(\boxed{320 \text{ million dollars}}\)[/tex].

### Part b: Find [tex]\( R(2) \)[/tex]
To find the revenue after 2 years, substituting [tex]\( x = 2 \)[/tex] into the function [tex]\( R(x) \)[/tex]:

[tex]\[ R(2) = \frac{1600 \cdot 2^2}{2^2 + 4} \][/tex]
[tex]\[ = \frac{1600 \cdot 4}{4 + 4} \][/tex]
[tex]\[ = \frac{6400}{8} \][/tex]
[tex]\[ = 800 \][/tex]

So, the revenue after 2 years is [tex]\(\boxed{800 \text{ million dollars}}\)[/tex].

### Part c: Find [tex]\( R(5) \)[/tex]
To find the revenue after 5 years, substituting [tex]\( x = 5 \)[/tex] into the function [tex]\( R(x) \)[/tex]:

[tex]\[ R(5) = \frac{1600 \cdot 5^2}{5^2 + 4} \][/tex]
[tex]\[ = \frac{1600 \cdot 25}{25 + 4} \][/tex]
[tex]\[ = \frac{40000}{29} \][/tex]
[tex]\[ \approx 1379.31 \][/tex]

So, the revenue after 5 years is approximately [tex]\(\boxed{1379.31 \text{ million dollars}}\)[/tex].

### Part d: Find [tex]\( R(10) \)[/tex]
To find the revenue after 10 years, substituting [tex]\( x = 10 \)[/tex] into the function [tex]\( R(x) \)[/tex]:

[tex]\[ R(10) = \frac{1600 \cdot 10^2}{10^2 + 4} \][/tex]
[tex]\[ = \frac{1600 \cdot 100}{100 + 4} \][/tex]
[tex]\[ = \frac{160000}{104} \][/tex]
[tex]\[ \approx 1538.46 \][/tex]

So, the revenue after 10 years is approximately [tex]\(\boxed{1538.46 \text{ million dollars}}\)[/tex].

### Summary
- After 1 year, the revenue is 320 million dollars.
- After 2 years, the revenue is 800 million dollars.
- After 5 years, the revenue is approximately 1379.31 million dollars.
- After 10 years, the revenue is approximately 1538.46 million dollars.

By evaluating the function at the specified values, we get a clear view of the total revenue from the sale of the book over a period of years.
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