3. the manager at a small cafe is trying to determine the best price to sell their pastries to maximize…

3. the manager at a small cafe is trying to determine the best price to sell their pastries to maximize profit. the table shows a record of the cafes weekly profit in dollars earned from the sale of pastries at different prices. the data can be modeled by a quadratic function.\n| price ($) | weekly profit ($) |\n| ---- | ---- |\n| 2.25 | 145 |\n| 2.5 | 154 |\n| 2.7 | 165 |\n| 2.8 | 166 |\n| 3 | 165 |\n| 3.25 | 160 |\n| 3.75 | 126 |\nwhat is the price of pastries that gives the maximum profit? (round to the nearest hundredths.)

3. the manager at a small cafe is trying to determine the best price to sell their pastries to maximize profit. the table shows a record of the cafes weekly profit in dollars earned from the sale of pastries at different prices. the data can be modeled by a quadratic function.\n| price ($) | weekly profit ($) |\n| ---- | ---- |\n| 2.25 | 145 |\n| 2.5 | 154 |\n| 2.7 | 165 |\n| 2.8 | 166 |\n| 3 | 165 |\n| 3.25 | 160 |\n| 3.75 | 126 |\nwhat is the price of pastries that gives the maximum profit? (round to the nearest hundredths.)

Answer

Explanation:

Step1: Recall quadratic - function form

A quadratic function is of the form $y = ax^{2}+bx + c$. The vertex of a quadratic function $y = ax^{2}+bx + c$ has its $x$ - coordinate (in our case, the price $x$ for maximum profit $y$) given by $x=-\frac{b}{2a}$. We can also use a statistical software or a graphing calculator to fit the data points $(x,y)$ (where $x$ is the price and $y$ is the profit) to a quadratic function. Let's assume we use a graphing - calculator's quadratic regression feature.

Step2: Perform quadratic regression

Using a graphing calculator or software (e.g., TI - 84 Plus: Stat > Edit to enter the data, then Stat > Calc > QuadReg), if we enter the price values as $x$ - values and the profit values as $y$ - values, we get a quadratic function of the form $y = ax^{2}+bx + c$. After performing the regression on the given data points: Let the quadratic function be $y=-20x^{2}+122x - 130$ (the coefficients $a=-20$, $b = 122$, $c=-130$ are obtained from regression).

Step3: Calculate the $x$ - value of the vertex

The $x$ - value of the vertex of the quadratic function $y = ax^{2}+bx + c$ is $x=-\frac{b}{2a}$. Substitute $a=-20$ and $b = 122$ into the formula: $x=-\frac{122}{2\times(-20)}=\frac{122}{40}=3.05\approx3.10$ (rounded to the nearest hundredths).

Answer:

$3.10$