The total weekly revenue earned at Royal Ruby Retailers is given by
R(p) = −
7
3
p2 + 154p
where p is the price (in dollars) RRR charges per ruby. Use this function to determine the following.
(a)
Determine the weekly revenue, to the nearest dollar, when the price is set at $20/ruby.
$
(b)
Determine the weekly revenue, to the nearest dollar, when the price is set at $200/ruby.
$
Interpret your result.
The domain of the demand function includes p = $200.
The range of the demand function includes p = $200.
The range of the demand function cannot realistically include p = $200.
The domain of the demand function cannot realistically include p = $200.
(c)
Determine the price RRR should charge in order to obtain a weekly revenue of $2,541.

### (a) Weekly revenue when the price is set at $20 per ruby:
The revenue is approximately **$2,147**.

### (b) Weekly revenue when the price is set at $200 per ruby:
The revenue is approximately **-$62,533**.

#### Interpretation:
The negative revenue when the price is set at $200 suggests that at this price point, the model predicts a significant loss for Royal Ruby Retailers. This implies that setting such a high price is not realistic within the function's practical constraints. 

- **Domain** of the function likely includes $ p = 200 $, but charging this price leads to a revenue loss.
- **Range** of the function cannot realistically include such a large negative revenue at high prices, suggesting the demand decreases significantly.

Now, I'll proceed to part (c), determining the price that would result in a weekly revenue of $2,541.

It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to guide you through the process of solving part (c) manually or assist with anything else?

Here are five questions that could expand on this:

1. How does the price affect the revenue in general based on the given function?
2. What does the domain of a function represent in real-world terms for businesses?
3. Could you graph the revenue function to visualize how price changes affect revenue?
4. How can Royal Ruby Retailers determine the best price to maximize revenue using calculus?
5. What is the significance of negative revenue in this context?

**Tip:** When solving quadratic equations like this, it's useful to check for both solutions, as one may not be practical for the real-world scenario.

The total weekly revenue earned at Royal Ruby Retailers is given by R(p) = −(7/3)p^2 + 154p where p is the price (in dollars) RRR charges per ruby. Use this function to determine the following: (a) Determine the weekly revenue, to the nearest dollar, when the price is set at $20/ruby. (b) Determine the weekly revenue, to the nearest dollar, when the price is set at $200/ruby. Interpret your result. (c) Determine the price RRR should charge in order to obtain a weekly revenue of $2,541.

Explore how to calculate the weekly revenue for Royal Ruby Retailers using the quadratic revenue function R(p) = −(7/3)p^2 + 154p. Learn how different price points like $20 and $200 affect revenue, and determine the optimal price for a target revenue of $2,541. This problem demonstrates real-world applications of quadratic functions and is suitable for high school students in Grades 10-12.

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