**Pierce Manufacturing's Revenue Analysis**Pierce Manufacturing determines that the daily revenue, in dollars, from the sale of $ x $ lawn chairs is represented by the equation:\[ R(x) = 0.007x^3 + 0.02x^2 + 4x. \]Currently, Pierce sells 65 lawn chairs daily. The following questions address the calculation of daily revenue, marginal revenue, and the implications of a change in sales.**a) Current Daily Revenue**- *Question*: What is the current daily revenue?- *Explanation*: Substitute $ x = 65 $ into the revenue equation $ R(x) $ to find the current daily revenue.**b) Equation for Marginal Revenue**- *Question*: What is the equation for marginal revenue?- *Explanation*: Marginal revenue is found by taking the derivative of the revenue function $ R(x) $.**c) Marginal Revenue when $ x = 65 $**- *Question*: What is the marginal revenue when $ x = 65 $?- *Explanation*: Substitute $ x = 65 $ into the marginal revenue equation derived in part b to find the specific marginal revenue for 65 lawn chairs.**d) Weekly Revenue Estimation with Increased Sales**- *Question*: Use your answer from part c to estimate the weekly revenue if sales increase to 66 lawn chairs daily.- *Explanation*: The obtained marginal revenue for 65 chairs is used to estimate additional revenue per day when selling 66 chairs. Multiply this by the number of days in a week to find the total change in weekly revenue.Each part of this analysis builds on calculus concepts related to finding revenue functions, derivatives, and using marginal analysis for decision-making.

Answered: Pierce Manufacturing determines that…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Pierce Manufacturing's Revenue Analysis**

Pierce Manufacturing determines that the daily revenue, in dollars, from the sale of $ x $ lawn chairs is represented by the equation:

\[ R(x) = 0.007x^3 + 0.02x^2 + 4x. \]

Currently, Pierce sells 65 lawn chairs daily. The following questions address the calculation of daily revenue, marginal revenue, and the implications of a change in sales.

**a) Current Daily Revenue**

- *Question*: What is the current daily revenue?
- *Explanation*: Substitute $ x = 65 $ into the revenue equation $ R(x) $ to find the current daily revenue.

**b) Equation for Marginal Revenue**

- *Question*: What is the equation for marginal revenue?
- *Explanation*: Marginal revenue is found by taking the derivative of the revenue function $ R(x) $.

**c) Marginal Revenue when $ x = 65 $**

- *Question*: What is the marginal revenue when $ x = 65 $?
- *Explanation*: Substitute $ x = 65 $ into the marginal revenue equation derived in part b to find the specific marginal revenue for 65 lawn chairs.

**d) Weekly Revenue Estimation with Increased Sales**

- *Question*: Use your answer from part c to estimate the weekly revenue if sales increase to 66 lawn chairs daily.
- *Explanation*: The obtained marginal revenue for 65 chairs is used to estimate additional revenue per day when selling 66 chairs. Multiply this by the number of days in a week to find the total change in weekly revenue.

Each part of this analysis builds on calculus concepts related to finding revenue functions, derivatives, and using marginal analysis for decision-making.

Expert Solution

Step 1: Introduction

Since you have posted a question with multiple subparts, we will solve only the first three subparts of the question. Kindly repost the others in the next question.

Given:

The daily revenue from the sale of $x$ lawn chairs is given by the function $R open parentheses x close parentheses equals 0.007 x cubed plus 0.02 x squared plus 0.4 x$ , where the daily revenue is in dollars.

Also, given that currently, the total quantity of lawn chairs sold daily by Pierce is $65$ .

To Find:

We have to determine,

a) the current daily revenue

b) the equation for marginal revenue

c) the marginal revenue when $x equals 65$