Your company manufactures two models of speakers, the Ultra Mini and the Big Stack. Demand for each depends partly on the price of the other. If one is expensive, then more people will buy the other. If p1 is the price of the Ultra Mini, and p2 is the price of the Big Stack, demand (quantity sold) for the Ultra Mini is given by
q1 = 100,000 − 400p1 + 10p2
where q1 represents the number of Ultra Minis that will be sold in a year. The demand (quantity sold) for the Big Stack is given by
q2 = 150,000 + 10p1 − 400p2.
Find the prices for the Ultra Mini and the Big Stack that will maximize your total revenue. Remember that revenue for each item is price*quantity. (Round your answers to two decimal places.)

We are given the demand functions for two models of speakers: the Ultra Mini and the Big Stack. The quantities sold, \( q_1 \) and \( q_2 \), depend on the prices \( p_1 \) and \( p_2 \). To maximize total revenue, we need to express total revenue as a function of the prices and then find the optimal prices by maximizing that function.

### Given demand functions:
- \( q_1 = 100,000 - 400p_1 + 10p_2 \) (quantity of Ultra Minis sold)
- \( q_2 = 150,000 + 10p_1 - 400p_2 \) (quantity of Big Stacks sold)

### Revenue functions:
The revenue for each product is the price multiplied by the quantity sold.

- Revenue from Ultra Mini: 
  \[
  R_1 = p_1 \cdot q_1 = p_1 \cdot (100,000 - 400p_1 + 10p_2)
  \]
- Revenue from Big Stack: 
  \[
  R_2 = p_2 \cdot q_2 = p_2 \cdot (150,000 + 10p_1 - 400p_2)
  \]

The total revenue \( R \) is the sum of the revenues from both products:
\[
R = R_1 + R_2 = p_1 \cdot (100,000 - 400p_1 + 10p_2) + p_2 \cdot (150,000 + 10p_1 - 400p_2)
\]

Now, we need to maximize this total revenue function by finding the optimal values of \( p_1 \) and \( p_2 \). To do this, we will first expand the expression for \( R \), then take partial derivatives with respect to \( p_1 \) and \( p_2 \), set them equal to zero, and solve for the prices.

Let's calculate.The optimal prices for maximizing total revenue are:

- \( p_1 = \frac{207500}{1599} \approx 129.80 \) (price of the Ultra Mini)
- \( p_2 = \frac{305000}{1599} \approx 190.68 \) (price of the Big Stack)

These are the prices at which your company will maximize its total revenue.

Would you like further details on how this was calculated or any additional questions?

Here are some related questions:
1. How does changing the price of one model affect the demand for the other?
2. What would happen to total revenue if both prices were increased by 10%?
3. How can we incorporate cost functions to find profit-maximizing prices instead?
4. How does the elasticity of demand impact revenue in this case?
5. Can these demand functions be adjusted to account for market saturation?

**Tip:** Always check for local maximums by confirming the second derivative tests or analyzing concavity when maximizing revenue functions.

Your company manufactures two models of speakers, the Ultra Mini and the Big Stack. Demand for each depends partly on the price of the other. If p1 is the price of the Ultra Mini, and p2 is the price of the Big Stack, demand (quantity sold) for the Ultra Mini is given by q1 = 100,000 − 400p1 + 10p2 where q1 represents the number of Ultra Minis that will be sold in a year. The demand (quantity sold) for the Big Stack is given by q2 = 150,000 + 10p1 − 400p2. Find the prices for the Ultra Mini and the Big Stack that will maximize your total revenue. Remember that revenue for each item is price*quantity. (Round your answers to two decimal places.)

Discover how to calculate the optimal prices for maximizing total revenue in the manufacturing of Ultra Mini and Big Stack speakers. This problem involves demand functions and revenue calculations, applying concepts such as partial derivatives and optimization. Detailed steps included for college-level learners.

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