Part B: The smoothie chain makes multiple $0.15 increases to the average prices of their smoothies. The table shows the average profit of the chain compared to the number of price increases. The data models a quadratic function.
| Number of Price Increases (x) | Profit (millions) (y) |
| :---------------------------- | :-------------------- |
| 25 | 204 |
| 60 | 285 |
| 105 | 368 |
| 132 | 384 |
| 155 | 376 |
| 172 | 356 |
| 200 | 318 |
| 254 | 185 |
| 290 | 70 |
Use the data in the table to answer questions 3-5.
3. Use technology or hand calculations to determine the equation for the quadratic function modeled by the data in the table. Show an image of your final answer. (10 points)
4. Using the equation from question 3, determine the maximum profit. (5 points)
5. Using the equation from question 3, determine how many price increases will cause the smoothie chain to have zero profit. (7 points)
Answer (2)
The quadratic function is modeled as y = − 0.01356 x 2 + 3.74276 x + 117.91363 , with a maximum profit of about $376.09 million at approximately 138 price increases. Additionally, around 304 price increases will lead to zero profit.
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Determine the quadratic function: Fit the data to the quadratic form y = a x 2 + b x + c , resulting in y = − 0.01356 x 2 + 3.74276 x + 117.91363 .
Calculate the number of price increases for maximum profit: Use x = 2 a − b to find x ≈ 138 .
Determine the maximum profit: Substitute x ≈ 138 into the quadratic equation to find y ≈ 376.09 .
Find the number of price increases for zero profit: Solve the quadratic equation for y = 0 using the quadratic formula, resulting in x ≈ 304 .
The equation for the quadratic function is y = − 0.01356 x 2 + 3.74276 x + 117.91363 . The maximum profit is approximately $376.09 million, which occurs at approximately 138 price increases. The number of price increases that will cause the smoothie chain to have zero profit is approximately 304 .
Explanation
Problem Analysis We are given a table of data relating the number of price increases ( x ) of smoothies to the profit ( y ) of a smoothie chain. The data is modeled by a quadratic function, and we need to find the equation of this quadratic function, the maximum profit, and the number of price increases that result in zero profit.
Finding the Quadratic Equation First, we need to determine the equation of the quadratic function that models the data. A quadratic function has the form y = a x 2 + b x + c . Using the data points from the table, we can use technology to find the coefficients a , b , and c . By fitting a quadratic polynomial to the data, we find the coefficients to be approximately:
a = − 0.01356 b = 3.74276 c = 117.91363
Therefore, the quadratic function is:
y = − 0.01356 x 2 + 3.74276 x + 117.91363
Determining Maximum Profit Next, we need to find the maximum profit. The x-value that maximizes the profit can be found using the formula x = 2 a − b . Substituting the values of a and b we found earlier:
x = 2 ×− 0.01356 − 3.74276 ≈ 137.96
This means that the maximum profit occurs at approximately 138 price increases. To find the maximum profit, we substitute this value of x back into the quadratic equation:
y = − 0.01356 ( 137.96 ) 2 + 3.74276 ( 137.96 ) + 117.91363 ≈ 376.09
Therefore, the maximum profit is approximately $376.09 million.
Finding the Number of Price Increases for Zero Profit Finally, we need to determine the number of price increases that will result in zero profit. This means we need to solve the quadratic equation − 0.01356 x 2 + 3.74276 x + 117.91363 = 0 for x . We can use the quadratic formula:
x = 2 a − b ± b 2 − 4 a c
Substituting the values of a , b , and c :
x = 2 ( − 0.01356 ) − 3.74276 ± ( 3.74276 ) 2 − 4 ( − 0.01356 ) ( 117.91363 )
x = ( − 0.02712 ) − 3.74276 ± 13.9902 − ( − 6.400 )
x = − 0.02712 − 3.74276 ± 20.3902
x = − 0.02712 − 3.74276 ± 4.51555
This gives us two possible values for x :
x 1 = − 0.02712 − 3.74276 − 4.51555 ≈ 304.47 x 2 = − 0.02712 − 3.74276 + 4.51555 ≈ − 28.55
Since the number of price increases cannot be negative, we discard the negative solution. Therefore, the number of price increases that will result in zero profit is approximately 304.
Final Answer The equation for the quadratic function is y = − 0.01356 x 2 + 3.74276 x + 117.91363 . The maximum profit is approximately $376.09 million, which occurs at approximately 138 price increases. The number of price increases that will cause the smoothie chain to have zero profit is approximately 304.
Examples
Understanding quadratic functions can help businesses model their profits based on various factors, such as price changes. For example, a store owner might want to determine the optimal price point for a product to maximize their profit. By collecting data on sales and prices, they can create a quadratic model to predict how changes in price will affect their profit. This allows them to make informed decisions about pricing strategies, promotions, and inventory management, ultimately leading to increased profitability. Similarly, manufacturers can use quadratic models to optimize production processes, minimizing costs and maximizing output.
