Math Problem Statement

The revenue for a manufacturer is​ R(x) thousand​ dollars, where x is the number of units of goods produced​ (and sold) and R and Upper R prime are the functions given in the graphs on the right. 0 10 20 30 40 50 0 10 20 30 40 50 60 70 80 90 x y y equals Upper R left parenthesis x right parenthesis

A coordinate system has a horizontal x-axis labeled from 0 to 50 in increments of 2.5 and a vertical y-axis labeled from 0 to 90 in increments of 10. From left to right, a curve titled y equals Upper R left parenthesis x right parenthesis rises at a decreasing rate passing through the point (5, 17.9) and (27.5, 75.4) until it reaches the point (33, 78), and then falls at an increasing rate. All coordinates are approximate. 10 20 30 40 50 -3.6 -2.4 -1.2 0 1.2 2.4 3.6 x y y equals Upper R prime left parenthesis x right parenthesis

A coordinate system has a horizontal x-axis labeled from 0 to 50 in increments of 2.5 and a vertical y-axis labeled from negative 4 to 4 in increments of 0.4. From left to right, a curve titled y equals Upper R prime left parenthesis x right parenthesis falls at an increasing rate passing through (17.5, 2.6), (30.6, 0.4), and (33, 0). All coordinates are approximate. Question content area bottom Part 1 ​(a) What is the revenue from producing 27.5 units of​ goods? The revenue from producing 27.5 units of goods is approximately ​$    enter your response here. ​(Type an integer or a​ decimal.)

Solution

To find the revenue from producing 27.5 units of goods, we refer directly to the graph of R(x)R(x). According to the given description:

  • The graph of R(x)R(x) (revenue in thousand dollars) shows that when x=27.5x = 27.5, the corresponding y-value (revenue) is approximately 75.4.

Thus:

R(27.5)≈75.4 thousand dollarsR(27.5) \approx 75.4 \text{ thousand dollars}

So, the revenue from producing 27.5 units of goods is approximately:

$75,400\text{\$75,400}

Would you like further details or have any questions?

Related Questions

  1. What is the revenue from producing 5 units of goods based on the graph?
  2. How would you interpret the behavior of R′(x)R'(x) when x=33x = 33?
  3. At what production level does R′(x)R'(x) equal zero, and what does this indicate about revenue at that point?
  4. How can you use R′(x)R'(x) to determine the production level that maximizes revenue?
  5. If R′(x)<0R'(x) < 0 after a certain point, what does it indicate about revenue trends beyond that point?

Tip: The derivative graph R′(x)R'(x) gives important information about where the revenue is increasing or decreasing and where it reaches its maximum value.

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Math Problem Analysis

Mathematical Concepts

Calculus
Revenue Function
Derivative Analysis
Graph Interpretation

Formulas

R(x): Revenue function
R'(x): Derivative of revenue function

Theorems

Maxima and Minima of a function
Interpretation of the first derivative

Suitable Grade Level

Grades 11-12

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