Question
Question
PJ owns a small bakery where fresh bread is baked and sold daily. The profit function p(x)=−3x2+150x
models the bakery’s profits, in dollars, as a function of the cost per dozen of mini bread loaves. If PJ wants to earn over $1,800.00 in profits, then what does the price range of the mini bread loaves (per dozen) need to be?(1 point)
Responses
The bread should be priced either less than $4.38 or greater than $45.62 to make a profit of $1,800.00.
The bread should be priced either less than $4.38 or greater than $45.62 to make a profit of $1,800.00.
The bread should be priced greater than $0.00 and less than $60.00 to make a profit of $1,800.00.
The bread should be priced greater than $0.00 and less than $60.00 to make a profit of $1,800.00.
The bread should be priced between $20.00 and $30.00 to make a profit of $1,800.00.
The bread should be priced between $20.00 and $30.00 to make a profit of $1,800.00.
The bread should be priced either less than $20.00 or greater than $30.00 to make a profit of $1,800.00.
The bread should be priced either less than $20.00 or greater than $30.00 to make a profit of $1,800.00.
PJ owns a small bakery where fresh bread is baked and sold daily. The profit function p(x)=−3x2+150x
models the bakery’s profits, in dollars, as a function of the cost per dozen of mini bread loaves. If PJ wants to earn over $1,800.00 in profits, then what does the price range of the mini bread loaves (per dozen) need to be?(1 point)
Responses
The bread should be priced either less than $4.38 or greater than $45.62 to make a profit of $1,800.00.
The bread should be priced either less than $4.38 or greater than $45.62 to make a profit of $1,800.00.
The bread should be priced greater than $0.00 and less than $60.00 to make a profit of $1,800.00.
The bread should be priced greater than $0.00 and less than $60.00 to make a profit of $1,800.00.
The bread should be priced between $20.00 and $30.00 to make a profit of $1,800.00.
The bread should be priced between $20.00 and $30.00 to make a profit of $1,800.00.
The bread should be priced either less than $20.00 or greater than $30.00 to make a profit of $1,800.00.
The bread should be priced either less than $20.00 or greater than $30.00 to make a profit of $1,800.00.
Answers
GPT-4o mini
To determine the price range for mini bread loaves to achieve profits over $1,800, we need to solve the inequality based on the profit function given:
\[ p(x) = -3x^2 + 150x \]
We need to find where:
\[ p(x) > 1800 \]
First, we set up the equation:
\[ -3x^2 + 150x - 1800 > 0 \]
Next, we can solve the corresponding equation:
\[ -3x^2 + 150x - 1800 = 0 \]
To simplify the equation, we can divide all terms by -3:
\[ x^2 - 50x + 600 = 0 \]
Now, we can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -50, c = 600 \):
1. Calculate the discriminant:
\[ b^2 - 4ac = (-50)^2 - 4 \cdot 1 \cdot 600 = 2500 - 2400 = 100 \]
2. Now, use the quadratic formula:
\[ x = \frac{50 \pm \sqrt{100}}{2 \cdot 1} \]
\[ x = \frac{50 \pm 10}{2} \]
This gives us two solutions:
1. \( x = \frac{60}{2} = 30 \)
2. \( x = \frac{40}{2} = 20 \)
Now, our quadratic can be factored as:
\[ (x - 20)(x - 30) > 0 \]
The critical points are \( x = 20 \) and \( x = 30 \). We can test the intervals:
- For \( x < 20 \): Choose \( x = 0 \), \( (0-20)(0-30) = (−20)(−30) = 600 > 0 \)
- For \( 20 < x < 30 \): Choose \( x = 25 \), \( (25-20)(25-30) = (5)(-5) = -25 < 0 \)
- For \( x > 30 \): Choose \( x = 40 \), \( (40-20)(40-30) = (20)(10) = 200 > 0 \)
Thus, the solution to the inequality \( (x - 20)(x - 30) > 0 \) is:
\[ x < 20 \quad \text{or} \quad x > 30 \]
Therefore, PJ should price the mini bread loaves either less than $20.00 or greater than $30.00 to earn profits over $1,800.
Thus, the correct response is:
**The bread should be priced either less than $20.00 or greater than $30.00 to make a profit of $1,800.00.**
\[ p(x) = -3x^2 + 150x \]
We need to find where:
\[ p(x) > 1800 \]
First, we set up the equation:
\[ -3x^2 + 150x - 1800 > 0 \]
Next, we can solve the corresponding equation:
\[ -3x^2 + 150x - 1800 = 0 \]
To simplify the equation, we can divide all terms by -3:
\[ x^2 - 50x + 600 = 0 \]
Now, we can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -50, c = 600 \):
1. Calculate the discriminant:
\[ b^2 - 4ac = (-50)^2 - 4 \cdot 1 \cdot 600 = 2500 - 2400 = 100 \]
2. Now, use the quadratic formula:
\[ x = \frac{50 \pm \sqrt{100}}{2 \cdot 1} \]
\[ x = \frac{50 \pm 10}{2} \]
This gives us two solutions:
1. \( x = \frac{60}{2} = 30 \)
2. \( x = \frac{40}{2} = 20 \)
Now, our quadratic can be factored as:
\[ (x - 20)(x - 30) > 0 \]
The critical points are \( x = 20 \) and \( x = 30 \). We can test the intervals:
- For \( x < 20 \): Choose \( x = 0 \), \( (0-20)(0-30) = (−20)(−30) = 600 > 0 \)
- For \( 20 < x < 30 \): Choose \( x = 25 \), \( (25-20)(25-30) = (5)(-5) = -25 < 0 \)
- For \( x > 30 \): Choose \( x = 40 \), \( (40-20)(40-30) = (20)(10) = 200 > 0 \)
Thus, the solution to the inequality \( (x - 20)(x - 30) > 0 \) is:
\[ x < 20 \quad \text{or} \quad x > 30 \]
Therefore, PJ should price the mini bread loaves either less than $20.00 or greater than $30.00 to earn profits over $1,800.
Thus, the correct response is:
**The bread should be priced either less than $20.00 or greater than $30.00 to make a profit of $1,800.00.**