### Problem OverviewIn economics, functions that involve revenue, cost, and profit are used. For example, suppose that $ R(x) $ and $ C(x) $ denote the total revenue and the total cost, respectively, of producing a new grocery cart for a wholesaler. The difference $ P(x) = R(x) - C(x) $ represents the total profit for producing $ x $ carts. Given $ R(x) = 58x - 0.3x^2 $ and $ C(x) = 5x + 15 $, find each of the following:### Tasksa) $ P(x) = $ (Simplify your answer. Use integers or decimals for any numbers in the expression.)b) $ R(80), C(80), \text{ and } P(80) $c) Using a graphing calculator, graph the three functions in the viewing window $[0, 210, 0, 3200]$.### Solution Steps1. **Calculate $ P(x) $:** - Formulate $ P(x) $ from the given $ R(x) $ and $ C(x) $ equations. - Simplify: $ P(x) = (58x - 0.3x^2) - (5x + 15) $.2. **Calculate $ R(80), C(80), \text{ and } P(80) $:** - Substitute $ x = 80 $ into $ R(x), C(x), \text{ and } P(x) $ and compute the values.3. **Graphing:** - Use a graphing calculator to plot $ R(x), C(x), \text{ and } P(x) $. - Set the viewing window $[0, 210, 0, 3200]$ to ensure all functions are visible within these limits.### Graphs Description- **Revenue Function $ R(x) $:** A parabola opening downward, showing how revenue changes with the number of carts produced.- **Cost Function $ C(x) $:** A linear line, showing a straightforward increase in cost with each additional cart.- **Profit Function $ P(x) $:** Represents the profitability, potentially a complex curve indicating where profits peak or drop to zero based on production quantity.Ensure to utilize

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In economics, functions that involve revenue, cost, and profit are used. For example, suppose that R(x) and C(x) denote the total revenue and the total cost, respectively, of producing a new grocery cart for a wholesaler. Then the difference P(x) = R(x) - C(x) represents the total profit for producing x carts. Given R(x) = 58x-0.3x² and C(x) = 5x + 15, find each of the following a) P(x) b) R(80), C(80), and P(80) c) Using a graphing calculator, graph the three functions in the viewing window [0, 210, 0, 3200]. a) P(x) = (Simplify your answer. Use integers or decimals for any numbers in the expression.) REED Clear all Check answer

In economics, functions that involve revenue, cost, and profit are used. For example, suppose that R(x) and C(x) denote the total revenue and the total cost, respectively, of producing a new grocery cart for a wholesaler. Then the difference P(x) = R(x) - C(x) represents the total profit for producing x carts. Given R(x) = 58x-0.3x² and C(x) = 5x + 15, find each of the following a) P(x) b) R(80), C(80), and P(80) c) Using a graphing calculator, graph the three functions in the viewing window [0, 210, 0, 3200]. a) P(x) = (Simplify your answer. Use integers or decimals for any numbers in the expression.) REED Clear all Check answer

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

### Problem Overview

In economics, functions that involve revenue, cost, and profit are used. For example, suppose that $ R(x) $ and $ C(x) $ denote the total revenue and the total cost, respectively, of producing a new grocery cart for a wholesaler. The difference $ P(x) = R(x) - C(x) $ represents the total profit for producing $ x $ carts. Given $ R(x) = 58x - 0.3x^2 $ and $ C(x) = 5x + 15 $, find each of the following:

### Tasks

a) $ P(x) = $

(Simplify your answer. Use integers or decimals for any numbers in the expression.)

b) $ R(80), C(80), \text{ and } P(80) $

c) Using a graphing calculator, graph the three functions in the viewing window $[0, 210, 0, 3200]$.

### Solution Steps

1. **Calculate $ P(x) $:**

- Formulate $ P(x) $ from the given $ R(x) $ and $ C(x) $ equations.
- Simplify: $ P(x) = (58x - 0.3x^2) - (5x + 15) $.

2. **Calculate $ R(80), C(80), \text{ and } P(80) $:**

- Substitute $ x = 80 $ into $ R(x), C(x), \text{ and } P(x) $ and compute the values.

3. **Graphing:**

- Use a graphing calculator to plot $ R(x), C(x), \text{ and } P(x) $.
- Set the viewing window $[0, 210, 0, 3200]$ to ensure all functions are visible within these limits.

### Graphs Description

- **Revenue Function $ R(x) $:** A parabola opening downward, showing how revenue changes with the number of carts produced.
- **Cost Function $ C(x) $:** A linear line, showing a straightforward increase in cost with each additional cart.
- **Profit Function $ P(x) $:** Represents the profitability, potentially a complex curve indicating where profits peak or drop to zero based on production quantity.

Ensure to utilize

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