Amanda Fall is starting up a new house painting business, Fall Colors. She has been advertising in the local newspaper for several months. Based on inquiries and informal surveys of the local housing market, she anticipates that she will get painting jobs at the rate of four per week (Poisson distributed). Amanda has also determined that it will take a four-person team of painters an average of 0.7 week (exponentially distributed) for a typical painting job.
a. Determine the number of teams of painters Amanda needs to hire so that customers will have to wait no longer than 2 weeks to get their houses painted.
b. If the average price for a painting job is $1,700 and Amanda pays a team of painters $500 per week, will she make any money?
To determine the number of teams of painters Amanda needs to hire so that customers will have to wait no longer than 2 weeks to get their houses painted, we need to calculate the average number of painting jobs waiting to be done at any given time.
First, we need to find the average arrival rate of painting jobs. It is given that she anticipates getting jobs at a rate of 4 per week (Poisson distributed). Therefore, the arrival rate (λ) is 4.
Secondly, we need to find the average service rate of the painting jobs. It is given that a typical painting job takes a team of 4 painters an average of 0.7 weeks (exponentially distributed). Therefore, the service rate (μ) is 1/0.7 = 1.43.
Using the M/M/c queuing formula, we can calculate the average number of customers in the system (painting jobs waiting and being serviced) using the following formula:
Ls = λ / (c * (c - λ/μ)),
Where:
Ls = average number of customers in the system,
λ = arrival rate,
μ = service rate,
c = number of servers (teams of painters).
In this case, since we're looking at painting jobs waiting to be done, we need to consider "c" as the number of teams of painters.
Let's substitute the values into the formula:
Ls = 4 / (c * (c - 4/1.43))
To ensure that customers wait no longer than 2 weeks, we set a maximum limit on the average number of jobs waiting:
Ls ≤ 2.
Now, we can solve for "c" by rearranging the equation:
4 / (c * (c - 4/1.43)) ≤ 2
Simplifying the equation, we have:
c * (c - 4/1.43) ≥ 2
Multiplying through by 1.43 to clear the fraction:
1.43c^2 - 4c ≥ 2.86
Rearranging the equation, we have:
1.43c^2 - 4c - 2.86 ≥ 0
To solve this quadratic inequality, we can use various methods. However, it's easier to use a graphing calculator or software to find the values of "c" that satisfy the inequality.
Using a graphing calculator or software, we find that "c" should be greater than or equal to 7 (approximately 7.55).
Therefore, Amanda needs to hire a minimum of 7 teams of painters to ensure customers wait no longer than 2 weeks to get their houses painted.
Now let's move on to the second question:
To determine if Amanda will make any money, we need to calculate the expected revenue and subtract the cost of the painters.
First, we calculate the expected revenue per painting job:
Expected revenue per painting job = Average price for a painting job = $1,700.
Next, let's calculate the cost of the team of painters for a typical painting job:
Cost of a team of painters = $500 per week * 0.7 weeks = $350.
Then, we need to determine the average number of painting jobs completed per week, which we already calculated as 4.
Finally, we can calculate the expected revenue and subtract the cost:
Expected revenue = Expected revenue per painting job * Average number of painting jobs completed per week = $1,700 * 4 = $6,800.
Expected cost = Cost of a team of painters * Average number of painting jobs completed per week = $350 * 4 = $1,400.
Profit = Expected revenue - Expected cost = $6,800 - $1,400 = $5,400.
Based on these calculations, Amanda will make a profit of $5,400.