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. a. write a function to represent the profit manuel makes selling n backpacks. b. 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. c. how much is his profit if he sells 43 backpacks? explain. setup cost $5,250 finished backpack assembly cost $54/backpack revenue $159/backpack

manuel calculates the business costs and profits to produce n hiking backpacks. manuels profit is his revenue minus his expenses. a. write a function to represent the profit manuel makes selling n backpacks. b. 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. c. how much is his profit if he sells 43 backpacks? explain. setup cost $5,250 finished backpack assembly cost $54/backpack revenue $159/backpack

Answer

Explanation:

Step1: Define revenue and cost functions

Revenue $r(n)=159n$ (since $159$ per backpack and $n$ backpacks sold), total cost $c(n)=5250 + 54n$ (setup cost of $5250$ plus $54$ per - backpack assembly cost).

Step2: Derive profit function

Profit $p(n)=r(n)-c(n)=159n-(5250 + 54n)=159n - 5250-54n=105n - 5250$.

Step3: Determine domain

The number of backpacks $n$ must be a non - negative integer. Since revenue is between $4000$ and $30000$, we first find the number of backpacks for the lower and upper bounds of revenue. For $r(n)=4000$, $n=\frac{4000}{159}\approx25.16$ and for $r(n)=30000$, $n=\frac{30000}{159}\approx188.68$. So a reasonable domain for one year is $n\in{0,1,\cdots,188}$ (since $n$ is a non - negative integer). The function is discrete because the number of backpacks sold $n$ can only take on integer values.

Step4: Calculate profit for $n = 43$

Substitute $n = 43$ into the profit function $p(n)$: $p(43)=105\times43-5250=4515 - 5250=-735$. This means Manuel has a loss of $735$ when he sells 43 backpacks.

Answer:

a. $p(n)=105n - 5250$ b. Domain: $n\in{0,1,\cdots,188}$, the function is discrete because the number of backpacks sold is a non - negative integer. c. The profit when $n = 43$ is $- 735$ (a loss of $735$).