Math / Algebra

QuestionThe total revenue from the sale of a popular book is approximated by the rational function $R(x)=\frac{1000 x^{2}}{x^{2}+4}$ , where $x$ is the number of years since publication and $\mathrm{R}(\mathrm{x})$ is the total revenue in millions of dollars. Use this function to complete parts a through d. (a) Find the total revenue at the end of the first year. \\square$ million

Studdy Solution

STEP 1

What is this asking? How much money was made from book sales after one year? Watch out! The function gives revenue in millions of dollars, so don't forget that important detail at the end!

STEP 2

1. Substitute the Value
2. Calculate the Revenue

STEP 3

We're given this fancy function $R(x) = \frac{1000 \cdot x^2}{x^2 + 4}$ , where $x$ is the number of years since the book came out, and $R(x)$ is the total revenue in millions of dollars.
We want to find the revenue after one year, so we'll plug in $x = \textbf{1}$ into our function!
It's as simple as that!

STEP 4

Substituting $x = \textbf{1}$ into the function, we get: $R(\textbf{1}) = \frac{1000 \cdot \textbf{1}^2}{\textbf{1}^2 + 4}$

STEP 5

Let's carefully calculate the revenue.
First, we simplify the numerator: $1000 \cdot \textbf{1}^2 = 1000 \cdot \textbf{1} = \textbf{1000}$ So, our numerator is 1000.

STEP 6

Now, let's tackle the denominator: $\textbf{1}^2 + 4 = \textbf{1} + 4 = \textbf{5}$ Our denominator is 5.

STEP 7

Putting it all together, we have: $R(\textbf{1}) = \frac{\textbf{1000}}{\textbf{5}}$ Now, we divide 1000 by 5: $\frac{\textbf{1000}}{\textbf{5}} = \textbf{200}$

STEP 8

So, after one year, the total revenue is $\textbf{200}$ million dollars!
Don't forget those units - that's a lot of cash! $\$$ $\textbf{200}$ million!

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Studdy Solution

STEP 1

What is this asking? How much money was made from book sales after one year? Watch out! The function gives revenue in millions of dollars, so don't forget that important detail at the end!

STEP 2

1. Substitute the Value
2. Calculate the Revenue

STEP 3

STEP 4

Substituting $x = \textbf{1}$ into the function, we get: $R(\textbf{1}) = \frac{1000 \cdot \textbf{1}^2}{\textbf{1}^2 + 4}$

STEP 5

Let's carefully calculate the revenue.
First, we simplify the numerator: $1000 \cdot \textbf{1}^2 = 1000 \cdot \textbf{1} = \textbf{1000}$ So, our numerator is 1000.

STEP 6

Now, let's tackle the denominator: $\textbf{1}^2 + 4 = \textbf{1} + 4 = \textbf{5}$ Our denominator is 5.

STEP 7

Putting it all together, we have: $R(\textbf{1}) = \frac{\textbf{1000}}{\textbf{5}}$ Now, we divide 1000 by 5: $\frac{\textbf{1000}}{\textbf{5}} = \textbf{200}$

STEP 8

So, after one year, the total revenue is $\textbf{200}$ million dollars!
Don't forget those units - that's a lot of cash! $\$$ $\textbf{200}$ million!

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Studdy Solution

STEP 1

What is this asking? How much money was made from book sales after one year? Watch out! The function gives revenue in *millions* of dollars, so don't forget that important detail at the end!

STEP 2

1. Substitute the Value2. Calculate the Revenue

STEP 3

STEP 4

Substituting x=1x = \textbf{1}x=1 into the function, we get: R(1)=1000⋅1212+4R(\textbf{1}) = \frac{1000 \cdot \textbf{1}^2}{\textbf{1}^2 + 4}R(1)=12+41000⋅12​

STEP 5

Let's carefully calculate the revenue.First, we simplify the numerator: 1000⋅12=1000⋅1=10001000 \cdot \textbf{1}^2 = 1000 \cdot \textbf{1} = \textbf{1000}1000⋅12=1000⋅1=1000 So, our numerator is **1000**.

STEP 6

Now, let's tackle the denominator: 12+4=1+4=5\textbf{1}^2 + 4 = \textbf{1} + 4 = \textbf{5}12+4=1+4=5 Our denominator is **5**.

STEP 7

Putting it all together, we have: R(1)=10005R(\textbf{1}) = \frac{\textbf{1000}}{\textbf{5}}R(1)=51000​ Now, we divide **1000** by **5**: 10005=200\frac{\textbf{1000}}{\textbf{5}} = \textbf{200}51000​=200

STEP 8

So, after one year, the total revenue is 200\textbf{200}200 *million* dollars!Don't forget those units - that's a lot of cash! $\$$200\textbf{200}200 million!

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What is this asking? How much money was made from book sales after one year? Watch out! The function gives revenue in millions of dollars, so don't forget that important detail at the end!

1. Substitute the Value
2. Calculate the Revenue

Substituting $x = \textbf{1}$ into the function, we get: $R(\textbf{1}) = \frac{1000 \cdot \textbf{1}^2}{\textbf{1}^2 + 4}$

Let's carefully calculate the revenue.
First, we simplify the numerator: $1000 \cdot \textbf{1}^2 = 1000 \cdot \textbf{1} = \textbf{1000}$ So, our numerator is 1000.

Now, let's tackle the denominator: $\textbf{1}^2 + 4 = \textbf{1} + 4 = \textbf{5}$ Our denominator is 5.

Putting it all together, we have: $R(\textbf{1}) = \frac{\textbf{1000}}{\textbf{5}}$ Now, we divide 1000 by 5: $\frac{\textbf{1000}}{\textbf{5}} = \textbf{200}$

So, after one year, the total revenue is $\textbf{200}$ million dollars!
Don't forget those units - that's a lot of cash! $\$$ $\textbf{200}$ million!