### Understanding Revenue Change in Widget SalesA company selling widgets has observed that the number of items sold, $ x $, depends upon the price, $ p $, at which they're sold, following the equation:\[ x = \frac{90000}{(0.6p + 1)^2} \]Due to inflation and increasing health benefit costs, the company has been increasing the price by $0.06 per month. Your task is to determine the rate at which revenue is changing when the company is selling widgets at $8 each.To find this, use the following steps:1. **Identify Revenue Function:** - Revenue, $ R $, is given by $ R = x \times p $. - Substitute the given equation for $ x $: \[ R = \left(\frac{90000}{(0.6p + 1)^2}\right) \times p \]2. **Calculate the Rate of Change of Revenue:** - Differentiate $ R $ with respect to time (given that $ p $ is changing over time). - Given the price increase rate ($ \frac{dp}{dt} $), find $ \frac{dR}{dt} $ when $ p = 8 $.3. **Substitute Values and Solve:** - Determine $ \frac{dR}{dt} $ knowing $ \frac{dp}{dt} = 0.06 $.**Problem Statement Revisited:**Due to inflation and increasing health benefit costs, the company has been increasing the price by $0.06 per month. Find the rate at which revenue is changing when the company is selling widgets at $8 each.\[ \text{Revenue is decreasing by} \ \fbox{} \ \text{dollars per month} \]**Hint:**Give your answer as a positive value.

Firms produce goods and services and sell them in the market to maximize their level of profit…

A company selling widgets has found that the number of items sold, x, depends upon the price, p at 90000 which they're sold, according the equation z (0.6p + 1)? Due to inflation and increasing health benefit costs, the company has been increasing the price by $0.06 per month. Find the rate at which revenue is changing when the company is selling widgets at $8 each. Revenue is decreasing by dollars per month Hint: Give your answer as a positive value.

A company selling widgets has found that the number of items sold, x, depends upon the price, p at 90000 which they're sold, according the equation z (0.6p + 1)? Due to inflation and increasing health benefit costs, the company has been increasing the price by $0.06 per month. Find the rate at which revenue is changing when the company is selling widgets at $8 each. Revenue is decreasing by dollars per month Hint: Give your answer as a positive value.

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Question

### Understanding Revenue Change in Widget Sales

A company selling widgets has observed that the number of items sold, $ x $, depends upon the price, $ p $, at which they're sold, following the equation:

\[ x = \frac{90000}{(0.6p + 1)^2} \]

Due to inflation and increasing health benefit costs, the company has been increasing the price by $0.06 per month. Your task is to determine the rate at which revenue is changing when the company is selling widgets at $8 each.

To find this, use the following steps:

1. **Identify Revenue Function:**
- Revenue, $ R $, is given by $ R = x \times p $.
- Substitute the given equation for $ x $:
\[ R = \left(\frac{90000}{(0.6p + 1)^2}\right) \times p \]

2. **Calculate the Rate of Change of Revenue:**
- Differentiate $ R $ with respect to time (given that $ p $ is changing over time).
- Given the price increase rate ($ \frac{dp}{dt} $), find $ \frac{dR}{dt} $ when $ p = 8 $.

3. **Substitute Values and Solve:**
- Determine $ \frac{dR}{dt} $ knowing $ \frac{dp}{dt} = 0.06 $.

**Problem Statement Revisited:**

Due to inflation and increasing health benefit costs, the company has been increasing the price by $0.06 per month. Find the rate at which revenue is changing when the company is selling widgets at $8 each.

\[ \text{Revenue is decreasing by} \ \fbox{} \ \text{dollars per month} \]

**Hint:**
Give your answer as a positive value.

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