4.) at a given store, more salespeople on the floor of the store produces more profit, but this only happens…

4.) at a given store, more salespeople on the floor of the store produces more profit, but this only happens to an extent. after that point, profit will begin to decline due to too many staff and not enough customers. a store has found that its profit, $p$, in thousands of dollars, as a function of $s$ sales people, can be approximated by $p(s)= - 0.5s^{2}+7s + 144.5$. how many sales people should be employed to maximize profit? what is the maximum profit? 4.) salespeople: ________ maximum profit: ________

4.) at a given store, more salespeople on the floor of the store produces more profit, but this only happens to an extent. after that point, profit will begin to decline due to too many staff and not enough customers. a store has found that its profit, $p$, in thousands of dollars, as a function of $s$ sales people, can be approximated by $p(s)= - 0.5s^{2}+7s + 144.5$. how many sales people should be employed to maximize profit? what is the maximum profit? 4.) salespeople: ________ maximum profit: ________

Answer

Explanation:

Step1: Identify the function type

The profit function $P(s)= - 0.5s^{2}+7s + 144.5$ is a quadratic function in the form $y = ax^{2}+bx + c$, where $a=-0.5$, $b = 7$, and $c = 144.5$.

Step2: Find the vertex - x - coordinate

For a quadratic function $y=ax^{2}+bx + c$, the $x$ - coordinate of the vertex (which gives the value of $s$ for maximum profit here) is $s=-\frac{b}{2a}$. Substitute $a=-0.5$ and $b = 7$ into the formula: [s=-\frac{7}{2\times(-0.5)}=\frac{-7}{-1}=7]

Step3: Find the maximum profit

Substitute $s = 7$ into the profit function $P(s)$: [P(7)=-0.5\times(7)^{2}+7\times7 + 144.5] [=-0.5\times49+49 + 144.5] [=-24.5+49+144.5] [=169]

Answer:

Salespeople: 7 Maximum Profit: 169 (in thousands of dollars)