College

Your major online bookstore is in direct competition with Amazon.com, BN.com, and BooksAMillion.com. Your company's daily revenue in dollars is given by:

[tex]\[ R(x, y, z) = 10,000 - 0.01x - 0.02y - 0.01z + 0.00001yz \][/tex]

where [tex]x[/tex], [tex]y[/tex], and [tex]z[/tex] are the online daily revenues of Amazon.com, BN.com, and BooksAMillion.com, respectively.

(a) If, on a certain day, Amazon.com shows revenue of [tex]\$9,000[/tex], while BN.com and BooksAMillion.com each show [tex]\$3,000[/tex], what does the model predict for your company's revenue that day?

[tex]\[ \square \][/tex]

(b) If Amazon.com and BN.com each show daily revenue of [tex]\$3,000[/tex], give an equation showing how your daily revenue depends on that of BooksAMillion.com.

[tex]\[ R(z) = \square \][/tex]

Answer :

Sure! Let's go through each part of the question step-by-step.

### Part (a)

For this part, we need to calculate your company's revenue using the given formula:

[tex]\[ R(x, y, z) = 10,000 - 0.01x - 0.02y - 0.01z + 0.00001yz \][/tex]

Given values for this day:
- Amazon's revenue ([tex]\(x\)[/tex]) = \[tex]$9,000
- BN.com's revenue (\(y\)) = \$[/tex]3,000
- BooksAMillion.com's revenue ([tex]\(z\)[/tex]) = \[tex]$3,000

Now, substitute these values into the revenue formula:

\[ R(9,000, 3,000, 3,000) = 10,000 - 0.01 \times 9,000 - 0.02 \times 3,000 - 0.01 \times 3,000 + 0.00001 \times 3,000 \times 3,000 \]

Let's calculate each term:
- \( 0.01 \times 9,000 = 90 \)
- \( 0.02 \times 3,000 = 60 \)
- \( 0.01 \times 3,000 = 30 \)
- \( 0.00001 \times 3,000 \times 3,000 = 90 \)

Now substitute these back to get the revenue:

\[ R = 10,000 - 90 - 60 - 30 + 90 = 9,910 \]

So, on that day, your company's revenue is predicted to be \$[/tex]9,910.

### Part (b)

Here, we need to find an equation for your company's daily revenue that only depends on BooksAMillion.com's daily revenue, [tex]\(z\)[/tex], given that both Amazon.com and BN.com show revenues of \[tex]$3,000 each day.

Substituting these values (\(x = 3,000\) and \(y = 3,000\)) into the formula, we get:

\[ R(z) = 10,000 - 0.01 \times 3,000 - 0.02 \times 3,000 - 0.01z + 0.00001 \times 3,000 \times z \]

Calculating the constant parts:
- \( 0.01 \times 3,000 = 30 \)
- \( 0.02 \times 3,000 = 60 \)

The formula simplifies to:

\[ R(z) = 10,000 - 30 - 60 - 0.01z + 0.00001 \times 3,000 \times z \]

Simplifying further:

\[ R(z) = 9,910 - 0.01z + 0.03z \]

\[ R(z) = 9,910 + 0.02z \]

Thus, your company's daily revenue as a function of BooksAMillion.com's revenue is given by:

\[ R(z) = 9,910 + 0.02z \]

This equation shows how your revenue varies with changes in BooksAMillion.com's revenue, while the other two companies have static revenues of \$[/tex]3,000 each.