Question
A woodworker makes wooden checkerboards.
Her profit is a function of the price she charges. This graph shows her total profits, y, based on the sales price, x, of each checkerboard.
Identify any zeros of the function, and interpret what the zeros mean in terms of the situation.
王1600
- 1000|
Profit （$）
500
→
K
20
40|
60
↓
Selling price ($)
80
• Zeros: x = 15 and x = 70.
These are the prices at which she neither makes nor loses money.
• Zeros: x = 15 and x = 70.
These are the prices at which she makes the least and greatest amounts of profit.
O Zero: x = 42.5.
This is the selling price that will give her the greatest profit.
• Zero: y = 1512.5.
This is the maximum amount of profit she can make.
Her profit is a function of the price she charges. This graph shows her total profits, y, based on the sales price, x, of each checkerboard.
Identify any zeros of the function, and interpret what the zeros mean in terms of the situation.
王1600
- 1000|
Profit （$）
500
→
K
20
40|
60
↓
Selling price ($)
80
• Zeros: x = 15 and x = 70.
These are the prices at which she neither makes nor loses money.
• Zeros: x = 15 and x = 70.
These are the prices at which she makes the least and greatest amounts of profit.
O Zero: x = 42.5.
This is the selling price that will give her the greatest profit.
• Zero: y = 1512.5.
This is the maximum amount of profit she can make.
Answers
GPT-4o mini
The zeros of the function are points where the total profits (y) are zero. In the context of this woodworker's situation, zeros occur at specific selling prices (x) where the woodworker neither makes money nor incurs a loss.
From your information, the identified zeros are at:
- \( x = 15 \): This is the selling price at which the woodworker's total profit is zero. This means that if she sells the checkerboard for $15, she will break even; that is, her revenue from selling the checkerboard will cover her costs but will not provide any profit.
- \( x = 70 \): This is another selling price where the total profit is also zero. Selling the checkerboard for $70 also results in breaking even, meaning that at this price point, her revenue equals her costs.
The interpretation of these zeros is that they represent the points at which the woodworker's pricing strategy results in neither profit nor loss. Selling at these prices would not be beneficial to her financial goals.
It's important to note that the assertion regarding the zeros being the points where she makes the least and greatest amounts of profit is incorrect. The zeros are not profit-maximizing prices but rather breakeven points.
Additionally:
- \( x = 42.5 \) appears to be indicated as the point giving the greatest profit, which seems to represent the maximum profit level but is not a zero.
- The maximum profit amount, possibly mentioned as \( y = 1512.5 \), represents the highest profit achieved and is not directly related to the zeros of the function.
In summary, the zeros (x = 15 and x = 70) indicate the breakeven points, while the maximum profit indicator (x = 42.5) represents the optimal selling price for maximized profit.
From your information, the identified zeros are at:
- \( x = 15 \): This is the selling price at which the woodworker's total profit is zero. This means that if she sells the checkerboard for $15, she will break even; that is, her revenue from selling the checkerboard will cover her costs but will not provide any profit.
- \( x = 70 \): This is another selling price where the total profit is also zero. Selling the checkerboard for $70 also results in breaking even, meaning that at this price point, her revenue equals her costs.
The interpretation of these zeros is that they represent the points at which the woodworker's pricing strategy results in neither profit nor loss. Selling at these prices would not be beneficial to her financial goals.
It's important to note that the assertion regarding the zeros being the points where she makes the least and greatest amounts of profit is incorrect. The zeros are not profit-maximizing prices but rather breakeven points.
Additionally:
- \( x = 42.5 \) appears to be indicated as the point giving the greatest profit, which seems to represent the maximum profit level but is not a zero.
- The maximum profit amount, possibly mentioned as \( y = 1512.5 \), represents the highest profit achieved and is not directly related to the zeros of the function.
In summary, the zeros (x = 15 and x = 70) indicate the breakeven points, while the maximum profit indicator (x = 42.5) represents the optimal selling price for maximized profit.