Annual profit in thousands of dollars is given by the function, P(x) = -.1x^2 + 50x - 300, where x is the number of items sold , x >= 0.
1.describe the meaning of the number -.1 in the formula.
2.describe the meaning of the number -300 in the formula.
3.find the profit for 5 different values
4.will this profit function have a maximum, if so, what is it?
5.what steps should the company take to prepare for the answer to part 5.
The volume V of a sphere of diameter d. To find the volume (the output), take the cube of the diameter (the input), then multiply by ð and divide by 6.
1. The number -.1 in the formula represents the coefficient of the x^2 term. In this context, it indicates the rate at which the profit decreases as the number of items sold increases. More specifically, it signifies that for every additional item sold, the profit will decrease by 0.1 thousand dollars (or $100).
2. The number -300 in the formula represents the constant term. This value signifies the fixed costs or expenses that the company incurs regardless of the number of items sold. It could include costs such as rent, utilities, or salaries. In this case, it suggests that the company has fixed costs of $300 thousand dollars that need to be covered.
3. To find the profit for different values, you can substitute the given values of x into the profit function and calculate the corresponding profit. For example:
- P(0) = -.1(0)^2 + 50(0) - 300 = -300 thousand dollars (this represents the profit when no items are sold).
- P(1) = -.1(1)^2 + 50(1) - 300 = -250 thousand dollars.
- P(2) = -.1(2)^2 + 50(2) - 300 = -200 thousand dollars.
- P(3) = -.1(3)^2 + 50(3) - 300 = -150 thousand dollars.
- P(4) = -.1(4)^2 + 50(4) - 300 = -100 thousand dollars.
4. Yes, this profit function will have a maximum point since the coefficient of the x^2 term is negative (-.1). The profit function is a quadratic equation, and the parabola will have a maximum point. In this case, the maximum profit indicates the point where the profit is highest before it starts decreasing.
5. To prepare for finding the maximum profit, the company should consider taking the following steps:
- Analyze the profit function to determine the selling range that will result in a profit.
- Calculate the maximum point of the profit function by finding the vertex of the quadratic equation. The x-coordinate of the vertex (in this case, the number of items sold) will indicate the quantity at which the maximum profit occurs.
- In addition to finding the maximum profit, assess whether it is economically viable for the company. Consider factors like production capacity, market demand, and fixed costs.
- Implement strategies to optimize profit, such as pricing strategies, cost-cutting measures, or exploring new markets.
- Regularly monitor sales and adjust business strategies accordingly to maximize profits in the long run.