For a certain company, the relationship between revenue (in billions of dollars) and the number of full-time employees
(in thousands) can be approximated by .
(a) Would be expected to be positive, negative or zero? Why?
(b) Find . What are the units?
© Find when and , and interpret the answer.

Ask by Fernandez Byrd. in the United States

Mar 12,2025

Question

For a certain company, the relationship between revenue (in billions of dollars) and the number of full-time employees
(in thousands) can be approximated by .
(a) Would be expected to be positive, negative or zero? Why?
(b) Find . What are the units?
© Find when and , and interpret the answer.

Ask by Fernandez Byrd. in the United States

Mar 12,2025

UpStudy AI · Accepted Answer

To solve the problem, we will follow the steps outlined:

### Step 1: Extract Known Conditions
We have the relationship given by the equation:
$$
5000 R^{2} = n^{3}
$$
where $$ R $$ is in billions of dollars and $$ n $$ is in thousands of employees.

### Step 2: Analyze the Problem
#### (a) Sign of $$ \frac{d n}{d R} $$
To determine whether $$ \frac{d n}{d R} $$ is positive, negative, or zero, we can analyze the relationship between $$ R $$ and $$ n $$:
- As revenue $$ R $$ increases, we expect the number of employees $$ n $$ to also increase, since a company typically hires more employees as it generates more revenue.
- Therefore, we would expect $$ \frac{d n}{d R} $$ to be positive.

### Step 3: Find $$ \frac{d n}{d R} $$
To find $$ \frac{d n}{d R} $$, we will differentiate the equation $$ 5000 R^{2} = n^{3} $$ with respect to $$ R $$.

1. Differentiate both sides:
   $$
   \frac{d}{dR}(5000 R^{2}) = \frac{d}{dR}(n^{3})
   $$
   This gives:
   $$
   10000 R = 3n^{2} \frac{d n}{d R}
   $$

2. Solve for $$ \frac{d n}{d R} $$:
   $$
   \frac{d n}{d R} = \frac{10000 R}{3n^{2}}
   $$

### Step 4: Determine Units
- The units of $$ R $$ are billions of dollars, and the units of $$ n $$ are thousands of employees.
- Therefore, the units of $$ \frac{d n}{d R} $$ will be:
$$
\text{Units of } \frac{d n}{d R} = \frac{\text{thousands of employees}}{\text{billions of dollars}} = \text{thousands of employees per billion dollars}
$$

### Step 5: Calculate $$ \frac{d n}{d R} $$ when $$ R=105 $$ and $$ n=300 $$
Now we will substitute $$ R = 105 $$ and $$ n = 300 $$ into the expression for $$ \frac{d n}{d R} $$:
$$
\frac{d n}{d R} = \frac{10000 \cdot 105}{3 \cdot (300)^{2}}
$$

Calculating the denominator:
$$
(300)^{2} = 90000
$$
Thus,
$$
\frac{d n}{d R} = \frac{10000 \cdot 105}{3 \cdot 90000}
$$

Now, let's calculate this value.

### Step 6: Perform the Calculation
We will calculate $$ \frac{d n}{d R} $$ using the values we have.

$$
\frac{d n}{d R} = \frac{1050000}{270000} = \frac{105}{27} \approx 3.8889
$$

### Step 7: Interpret the Answer
The value $$ \frac{d n}{d R} \approx 3.89 $$ means that for every additional billion dollars in revenue, the company is expected to hire approximately 3.89 thousand (or about 3890) more employees.

### Final Summary
- (a) $$ \frac{d n}{d R} $$ is expected to be positive.
- (b) $$ \frac{d n}{d R} = \frac{10000 R}{3n^{2}} $$ with units of thousands of employees per billion dollars.
- (c) When $$ R=105 $$ and $$ n=300 $$, $$ \frac{d n}{d R} \approx 3.89 $$, indicating an increase in employment with increased revenue.
(a) $$ \frac{d n}{d R} $$ is expected to be positive because as revenue increases, the number of employees also increases. (b) $$ \frac{d n}{d R} = \frac{10000 R}{3n^{2}} $$ with units of thousands of employees per billion dollars. (c) When $$ R=105 $$ and $$ n=300 $$, $$ \frac{d n}{d R} \approx 3.89 $$, meaning the company hires approximately 3,890 more employees for every additional billion dollars in revenue.

For a certain company, the relationship between revenue (in billions of dollars) and the number of full-time employees
(in thousands) can be approximated by .
(a) Would be expected to be positive, negative or zero? Why?
(b) Find . What are the units?
© Find when and , and interpret the answer.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

(a) Would be expected to be positive, negative or zero? Why?

(b) Find . What are the units?

© Find when and , and interpret the answer.

preguntas relacionadas

Latest Calculus Questions

For a certain company, the relationship between revenue (in billions of dollars) and the number of full-time employees (in thousands) can be approximated by . (a) Would be expected to be positive, negative or zero? Why? (b) Find . What are the units? © Find when and , and interpret the answer.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

(a) Would be expected to be positive, negative or zero? Why?

(b) Find . What are the units?

© Find when and , and interpret the answer.

preguntas relacionadas

Latest Calculus Questions

For a certain company, the relationship between revenue (in billions of dollars) and the number of full-time employees
(in thousands) can be approximated by .
(a) Would be expected to be positive, negative or zero? Why?
(b) Find . What are the units?
© Find when and , and interpret the answer.