ousing Starts and Construction Jobs The president of a major housing construction firm reports that the number of construction jobs (in millions) created is given by
N(x) = 1.47x
where x denotes the number of housing starts. Suppose the number of housing starts in the next t months is expected to be
x(t) = 
8(t + 10)2
(t + 10)2 + 2(t + 15)2
million units. Find an expression for the number of jobs (in millions) created in the next t months.
N(t) = 
 
How many jobs (in millions) will have been created 3 months and 12 months from now? (Round your answers to two decimal places.)
3 months	
 million jobs
12 months	
 million jobs

To solve this problem, we need to find an expression for the number of construction jobs \( N(t) \) created in the next \( t \) months. This expression is based on the function \( N(x) = 1.47x \), where \( x \) represents the number of housing starts, and \( x(t) \) is given by:

\[
x(t) = \frac{8(t + 10)^2}{(t + 10)^2 + 2(t + 15)^2}
\]

### Step 1: Find the Expression for \( N(t) \)

The number of construction jobs \( N(t) \) created in the next \( t \) months is:

\[
N(t) = 1.47 \cdot x(t)
\]

Substitute the expression for \( x(t) \):

\[
N(t) = 1.47 \cdot \frac{8(t + 10)^2}{(t + 10)^2 + 2(t + 15)^2}
\]

### Step 2: Calculate the Number of Jobs Created After 3 Months and 12 Months

Now, we need to compute \( N(t) \) for \( t = 3 \) and \( t = 12 \).

1. **For \( t = 3 \):**

\[
N(3) = 1.47 \cdot \frac{8(3 + 10)^2}{(3 + 10)^2 + 2(3 + 15)^2}
\]

2. **For \( t = 12 \):**

\[
N(12) = 1.47 \cdot \frac{8(12 + 10)^2}{(12 + 10)^2 + 2(12 + 15)^2}
\]

Let's calculate these values.The number of construction jobs created will be:

- **3 months from now**: Approximately \(2.43\) million jobs.
- **12 months from now**: Approximately \(2.93\) million jobs.

Would you like to know more details or have any questions?

Here are 5 related questions to consider:
1. How does the expression for \( x(t) \) change if we adjust the parameters?
2. What would be the effect on the number of jobs created if housing starts doubled?
3. How can we visualize the relationship between \( t \) and \( N(t) \)?
4. What is the rate of change of jobs created over time?
5. Can we determine the maximum number of jobs created in a given period?

**Tip:** To understand these relationships better, try plotting the function \( N(t) \) over a range of months.

The president of a major housing construction firm reports that the number of construction jobs (in millions) created is given by N(x) = 1.47x, where x denotes the number of housing starts. Suppose the number of housing starts in the next t months is expected to be x(t) = 8(t + 10)^2 / ((t + 10)^2 + 2(t + 15)^2) million units. Find an expression for the number of jobs (in millions) created in the next t months. How many jobs (in millions) will have been created 3 months and 12 months from now? (Round your answers to two decimal places.)

Learn how to find the number of construction jobs created from housing starts using the formula N(x) = 1.47x and a given function for housing starts over time. Includes a detailed step-by-step solution for calculating jobs after 3 and 12 months. Suitable for high school and early college students.

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