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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.
### 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.
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