https://r2.erweima.ai/mathbot/image/1726108418968go7l5xyo.jpg

You've provided a problem where we need to calculate the utility of a manufacturer when production increases from 9 units to 16 units. The marginal revenue $ MR $ is given as $ 15q^{-\frac{1}{2}} $ dollars per unit, and the marginal cost $ MC $ is given as $ 0.9q $ dollars per unit. Initially, the utility at 9 units is $500.

To find the utility at 16 units, we can use the information that utility $ U(q) $ is the integral of profit, which is $ MR - MC $.

1. **Calculate the profit function, $ P(q) $:**
   \[
   P(q) = MR - MC = 15q^{-\frac{1}{2}} - 0.9q
   \]

2. **Integrate $ P(q) $ from 9 to 16 to find the change in utility:**
   \[
   \Delta U = \int_{9}^{16} (15q^{-\frac{1}{2}} - 0.9q) \, dq
   \]

3. **Add $ \Delta U $ to the initial utility at 9 units to find the utility at 16 units.**

Let's calculate $ \Delta U $ using the integration.The change in utility $ \Delta U $ from producing 9 units to 16 units is $-48.75$. To find the utility at 16 units, we add this change to the initial utility at 9 units, which was $500.

\[ U(16) = 500 + (-48.75) = 451.25 \]

Therefore, the utility of the manufacturer when the production level is 16 units is $451.25.

Do you need further details on this calculation or any other aspect of the problem?

Here are some related questions you might consider:

1. How does the utility change if the production is increased further beyond 16 units?
2. What is the break-even production level where marginal revenue equals marginal cost?
3. How does changing the rate of marginal cost affect the utility at a specific production level?
4. What would be the utility if marginal revenue had a different functional form?
5. Calculate the maximum production level at which utility is maximized.

**Tip:** Understanding the relationship between production levels, revenue, cost, and utility can help optimize business strategies and resource allocation.

Un fabricante estima que el ingreso marginal es 150q^{-1/2} dólares por unidad cuando el nivel de producción es q unidades, y el costo marginal correspondiente es 0.9q dólares por unidad. Suponga que la utilidad del fabricante es 500 cuando el nivel de producción es de 9 unidades. ¿Cuál es la utilidad del fabricante cuando el nivel de producción es de 16 unidades?

Learn how to calculate the utility of a manufacturer when production increases from 9 to 16 units, using marginal revenue and marginal cost functions. This problem involves key calculus concepts such as integration and profit maximization.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation