Gadget Profit
![[justify]Justin, an economic analyst at a gadget company, knows that making more gadgets might not always increase profits due to rising costs or market saturation. To plan better, he uses a formula [math]P\left(x\right)=\frac{ax}{bx+c}[/math] to model profits per gadget, where:[/justify][list][*][math]x[/math] is the number of gadgets produced, in thousands. [/*][*][math]a[/math] is the maximum possible profit per gadget, in thousands of pesos (PHP)[/*][*][math]b[/math] is the rate at which profit per gadget slows down as production increases, and[/*][*][math]c[/math] is the minimum number of gadgets needed before starting to profit, in thousands[/*][/list]
A dynamic graph is shown to model the changes in profit relative to these factors.](https://www.geogebra.org/resource/xmyxy3e6/Mlopk4uI9YLm57eO/material-xmyxy3e6.png)
Justin, an economic analyst at a gadget company, knows that making more gadgets might not always increase profits due to rising costs or market saturation. To plan better, he uses a formula to model profits per gadget, where:
- is the number of gadgets produced, in thousands.
- is the maximum possible profit per gadget, in thousands of pesos (PHP)
- is the rate at which profit per gadget slows down as production increases, and
- is the minimum number of gadgets needed before starting to profit, in thousands
Based on Justin’s research of the market for the year, the values for the profit model are set as follows: , , and . Given this model, what is the maximum profit per gadget that the company can achieve as they increase production indefinitely?
Assist Justin in determining the optimal production level to maximize profits. Use sliders to adjust the values of a, b, and c, and observe their impact on the profit graph. Select the statements below that correctly describe how changes in a, b, and c affect the profit per gadget:
Justin is analyzing how external market factors could affect the company's profitability using the profit model , with initial values set at , , and . Due to forecasted changes in the market, two scenarios are presented:
Scenario 1: A 10% increase in due to improved consumer demand.
Scenario 2: A 15% increase in due to heightened competition.
Calculate the maximum profit per gadget for each scenario as production volume increases. Based on your calculations, which statement best describes the impact on maximum profit and its potential implications for Justin's business strategy?