manuel calculates the business costs and profits to produce n hiking backpacks. manuels profit is his…

manuel calculates the business costs and profits to produce n hiking backpacks. manuels profit is his revenue minus his expenses.\na. write a function to represent the profit manuel makes selling n backpacks.\nb. graph the profit function. what is a reasonable domain for this function for one year if his revenue is between $4,000 and $30,000? is the function discrete or continuous? explain.\nc. how much is his profit if he sells 43 backpacks? explain.\nsetup cost: $5,250\nassembly cost: $54/backpack\nrevenue: $159/backpack\na. the function p(n)=105n - 5250 represents the profit manuel makes selling n backpacks (simplify your answer)\nb. if the function is continuous, graph it using a line. if the function is discrete, graph it using points at n - values that are multiples of 5 in order to keep the graph legible. choose the correct graph below.

manuel calculates the business costs and profits to produce n hiking backpacks. manuels profit is his revenue minus his expenses.\na. write a function to represent the profit manuel makes selling n backpacks.\nb. graph the profit function. what is a reasonable domain for this function for one year if his revenue is between $4,000 and $30,000? is the function discrete or continuous? explain.\nc. how much is his profit if he sells 43 backpacks? explain.\nsetup cost: $5,250\nassembly cost: $54/backpack\nrevenue: $159/backpack\na. the function p(n)=105n - 5250 represents the profit manuel makes selling n backpacks (simplify your answer)\nb. if the function is continuous, graph it using a line. if the function is discrete, graph it using points at n - values that are multiples of 5 in order to keep the graph legible. choose the correct graph below.

Answer

Explanation:

Step1: Find the profit - function

The revenue per backpack is $159 and the assembly cost per backpack is $54, and there is a setup cost of $5250. The profit function $p(n)$ is the revenue minus the total cost. The revenue from selling $n$ backpacks is $159n$, and the total cost is the sum of the setup cost ($5250$) and the assembly cost ($54n$). So, $p(n)=159n-(54n + 5250)=159n-54n-5250 = 105n-5250$.

Step2: Determine the domain and the type of the function

The revenue $R = 159n$ and $4000\leq R\leq30000$. So, $4000\leq159n\leq30000$. Dividing through by 159 gives $\frac{4000}{159}\leq n\leq\frac{30000}{159}$. $\frac{4000}{159}\approx25.16$ and $\frac{30000}{159}\approx188.68$. Since $n$ represents the number of backpacks, $n$ is a non - negative integer. The function is discrete because the number of backpacks sold ($n$) can only be whole numbers.

Step3: Calculate the profit when $n = 43$

Substitute $n = 43$ into the profit function $p(n)=105n-5250$. Then $p(43)=105\times43-5250=4515 - 5250=-735$.

Answer:

a. $p(n)=105n - 5250$ b. The domain is $n\in\mathbb{Z}, \ 26\leq n\leq188$ (where $\mathbb{Z}$ is the set of integers), and the function is discrete because the number of backpacks sold must be a whole number. c. The profit when $n = 43$ is $- 735$. This means that when 43 backpacks are sold, Manuel has a loss of $735$ since the cost ($5250 + 54\times43=5250+2322 = 7572$) is greater than the revenue ($159\times43 = 6837$).