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.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![### 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2a9572e1-cd0e-48a9-8ed8-4d34eb3e96b9%2F03c0ade8-bb72-466a-9e1f-e11e34f95681%2Fzi00rw_processed.png&w=3840&q=75)
Transcribed Image Text:### 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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