Use a graphing calculator to find how many widgets to make in order to maximize profit. The price (p) of one widget is given as a funtion of the number sold (n) as p=20e^[(-n^2)/40000]and the total cost (c) to make n widgets is given by C = 2n^2 + 2000/( n +10). Assume #made = # sold + n. Profit (p)= Revenue (np) - Cost (c)
To use a graphing calculator to find the number of widgets that will maximize profit, we will first need to determine the profit equation. The profit (p) can be calculated as the revenue (np) minus the cost (c).
Given that the price (p) of one widget is given as a function of the number sold (n) as p = 20e^(-n^2/40000), and the total cost (c) to make n widgets is given by C = 2n^2 + 2000/(n + 10), we can calculate the profit equation as follows:
Profit (p) = Revenue (np) - Cost (c)
Now let's substitute the given equations into the profit equation:
Profit (p) = np - c
Profit (p) = n * (20e^(-n^2/40000)) - (2n^2 + 2000/(n + 10))
To find the number of widgets that will maximize profit, we need to find the maximum value of the profit equation. We can do this by graphing the equation and finding the maximum point.
To plot the graph, you can use a graphing calculator or software like Desmos or WolframAlpha. Once you have the graph, locate the point where the profit is maximized by finding the highest peak.
After finding the x-coordinate of the maximum point on the graph, it represents the number of widgets (n) to make in order to maximize profit.
To use a graphing calculator to find the number of widgets to make in order to maximize profit, we need to follow these steps:
1. Enter the function for the price of one widget (p) as a function of the number sold (n): p = 20e^((-n^2)/40000).
2. Enter the function for the total cost (c) to make n widgets: C = 2n^2 + 2000/(n + 10).
3. Enter the function for profit (p) as the difference between revenue and cost: Profit = Revenue - Cost.
4. Set up the revenue function by multiplying the price (p) by the number sold (n): Revenue = p * n.
5. Set up the cost function by replacing n in the cost equation (C) with (#made - #sold): Cost = 2(#made - #sold)^2 + 2000/((#made - #sold) + 10).
6. Substitute the revenue and cost functions into the profit equation to get an expression in terms of (#made - #sold): Profit = (20e^((-n^2)/40000))n - (2(#made - #sold)^2 + 2000/((#made - #sold) + 10)).
7. Simplify the expression for profit to obtain it as a function of (#made - #sold).
8. Use the graphing calculator to graph the profit function and find the maximum point on the graph by using the calculator's maximum function or by visually inspecting the graph.
9. The x-coordinate of the maximum point on the graph represents the value for (#made - #sold), which gives the optimum number of widgets to make in order to maximize profit.
10. To find the specific values for (#made) and (#sold), add the value of (#made - #sold) to the value of (#sold).
That's it! By using a graphing calculator, you can find the number of widgets to make in order to maximize profit based on the given price and cost functions.