Mario Has A Business Selling Muffins. Let { X $}$ Be The Price Of A Muffin. The Profit { P $}$ For Mario's Business Is Given By:${ P(x) = -100x^2 + 350x - 150 }$Choose The Inequality That Shows The Business Will Make A

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Mario's Muffin Business: A Mathematical Analysis of Profit

Mario has a business selling muffins, and he wants to maximize his profit. The price of a muffin is denoted by { x $}$, and the profit { P $}$ for Mario's business is given by the quadratic function:

P(x)=−100x2+350x−150{ P(x) = -100x^2 + 350x - 150 }

In this article, we will analyze the profit function and determine the inequality that shows the business will make a profit.

The profit function { P(x) $}$ is a quadratic function, which means it can be written in the form:

P(x)=ax2+bx+c{ P(x) = ax^2 + bx + c }

where { a $}$, { b $}$, and { c $}$ are constants. In this case, we have:

P(x)=−100x2+350x−150{ P(x) = -100x^2 + 350x - 150 }

To understand the behavior of the profit function, we need to analyze its graph. The graph of a quadratic function is a parabola, which opens downward if { a $}$ is negative and upward if { a $}$ is positive.

To graph the profit function, we can use the following steps:

  1. Find the x-intercepts of the function by setting { P(x) $}$ equal to zero and solving for { x $}$.
  2. Find the y-intercept of the function by evaluating { P(x) $}$ at { x = 0 $}$.
  3. Use the x-intercepts and y-intercept to draw the graph of the function.

Finding the x-Intercepts

To find the x-intercepts of the function, we set { P(x) $}$ equal to zero and solve for { x $}$:

−100x2+350x−150=0{ -100x^2 + 350x - 150 = 0 }

We can solve this quadratic equation using the quadratic formula:

x=−b±b2−4ac2a{ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} }

where { a $}$, { b $}$, and { c $}$ are the coefficients of the quadratic equation.

Plugging in the values, we get:

x=−350±3502−4(−100)(−150)2(−100){ x = \frac{-350 \pm \sqrt{350^2 - 4(-100)(-150)}}{2(-100)} }

Simplifying, we get:

x=−350±122500−60000−200{ x = \frac{-350 \pm \sqrt{122500 - 60000}}{-200} }

x=−350±62600−200{ x = \frac{-350 \pm \sqrt{62600}}{-200} }

x=−350±250−200{ x = \frac{-350 \pm 250}{-200} }

Solving for the two values of { x $}$, we get:

x1=−350+250−200=−100−200=0.5{ x_1 = \frac{-350 + 250}{-200} = \frac{-100}{-200} = 0.5 }

x2=−350−250−200=−600−200=3{ x_2 = \frac{-350 - 250}{-200} = \frac{-600}{-200} = 3 }

Finding the y-Intercept

To find the y-intercept of the function, we evaluate { P(x) $}$ at { x = 0 $}$:

P(0)=−100(0)2+350(0)−150{ P(0) = -100(0)^2 + 350(0) - 150 }

P(0)=−150{ P(0) = -150 }

Graphing the Profit Function

Using the x-intercepts and y-intercept, we can draw the graph of the profit function. The graph is a parabola that opens downward, with a vertex at { x = 1.75 $}$.

To determine the inequality that shows the business will make a profit, we need to analyze the profit function. Since the graph of the profit function is a parabola that opens downward, the function is negative for all values of { x $}$ greater than the vertex.

Finding the Inequality

To find the inequality that shows the business will make a profit, we need to find the values of { x $}$ for which the profit function is positive. Since the graph of the profit function is a parabola that opens downward, the function is positive for all values of { x $}$ less than the vertex.

Therefore, the inequality that shows the business will make a profit is:

x<1.75{ x < 1.75 }

In this article, we analyzed the profit function of Mario's muffin business and determined the inequality that shows the business will make a profit. The profit function is a quadratic function that opens downward, and the business will make a profit for all values of { x $}$ less than the vertex.

The final answer is: x<1.75\boxed{x < 1.75}
Mario's Muffin Business: A Mathematical Analysis of Profit - Q&A

In our previous article, we analyzed the profit function of Mario's muffin business and determined the inequality that shows the business will make a profit. In this article, we will answer some frequently asked questions about the profit function and the business.

Q: What is the profit function of Mario's muffin business?

A: The profit function of Mario's muffin business is given by the quadratic function:

P(x)=−100x2+350x−150{ P(x) = -100x^2 + 350x - 150 }

Q: What is the vertex of the profit function?

A: The vertex of the profit function is at { x = 1.75 $}$.

Q: What is the inequality that shows the business will make a profit?

A: The inequality that shows the business will make a profit is:

x<1.75{ x < 1.75 }

Q: What is the maximum profit of the business?

A: The maximum profit of the business is { P(1.75) $}$, which is equal to { 175 $}$.

Q: What is the minimum price at which the business will make a profit?

A: The minimum price at which the business will make a profit is { x = 0 $}$, which is equal to { 0 $}$.

Q: What is the maximum price at which the business will make a profit?

A: The maximum price at which the business will make a profit is { x = 1.75 $}$, which is equal to { 1.75 $}$.

Q: How can Mario increase the profit of his business?

A: Mario can increase the profit of his business by increasing the price of his muffins, but only up to a certain point. If he increases the price too much, the business will start to lose money.

Q: What is the relationship between the price of the muffins and the profit of the business?

A: The price of the muffins and the profit of the business are related by the profit function. If the price of the muffins increases, the profit of the business will also increase, but only up to a certain point.

In this article, we answered some frequently asked questions about the profit function of Mario's muffin business and the business itself. We hope that this article has provided you with a better understanding of the profit function and the business.

The final answer is: x<1.75\boxed{x < 1.75}