Problem 86 - Algebra and Trigonometry

Question

Profit A small-appliance manufacturer finds that the profit \(P\) (in dollars) generated by producing \(x\) microwave ovens per week is given by the formula \(P=\frac{1}{10} X(300-x)\) provided that \(0 \leq x \leq 200 .\) How many ovens must be manufactured in a given week to generate a profit of \(\$ 1250 ?\)

Step-by-Step Solution

Answer

50 ovens must be manufactured to generate a profit of $1250.

Step 1: Understand the profit equation

The formula given for the profit is \(P = \frac{1}{10} x(300 - x)\). This represents a quadratic equation in terms of \(x\), where \(x\) is the number of microwave ovens produced.

Step 2: Set the profit equation equal to 1250

Since we need to find how many ovens must be manufactured to generate a profit of \$1250, we set the equation \(P = \frac{1}{10} x(300 - x)\) equal to 1250: \[1250 = \frac{1}{10} x(300 - x)\]

Step 3: Simplify the equation

Multiply both sides of the equation by 10 to eliminate the fraction:\[12500 = x(300 - x)\]This simplifies to:\[12500 = 300x - x^2\]

Step 4: Rearrange into a standard quadratic equation

Rearrange the equation into the standard quadratic form:\[x^2 - 300x + 12500 = 0\]

Step 5: Solve the quadratic equation

Use the quadratic formula to solve for \(x\). The quadratic formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]For our equation \(x^2 - 300x + 12500 = 0\), \(a = 1\), \(b = -300\), and \(c = 12500\).

Step 6: Calculate the discriminant

Calculate the discriminant using the formula \(b^2 - 4ac\):\[(-300)^2 - 4(1)(12500) = 90000 - 50000 = 40000\]

Step 7: Find the roots using the quadratic formula

Substitute the discriminant and other values into the quadratic formula:\[x = \frac{-(-300) \pm \sqrt{40000}}{2(1)}\]\[x = \frac{300 \pm 200}{2}\]This gives roots:\[x = \frac{300 + 200}{2} = 250\]\[x = \frac{300 - 200}{2} = 50\]

Step 8: Choose the valid solution

Since we are told that \(0 \leq x \leq 200\), the solution \(x = 250\) is not valid because it exceeds 200. Therefore, \(x = 50\) is the valid solution.

Key Concepts

Profit Equation

A profit equation is a mathematical formula that represents the profit a company makes based on a certain number of units produced or sold.
In this case, the equation given is \(P = \frac{1}{10} x(300 - x)\).
This formula specifies how much profit \(P\), in dollars, is generated by producing \(x\) microwave ovens per week. Here, the values of \(x\) must range between 0 and 200 because producing outside this range is not feasible or considered profitable for the company.The structure \(\frac{1}{10} x(300 - x)\) is a quadratic relationship. This means as the number of ovens \(x\) changes, it affects the profit in a curvilinear manner.
When you multiply \(x\) by \((300-x)\), it reflects how profit initially rises with increased production until it reaches an optimal point, and then starts decreasing again if production continues to increase.

Quadratic Formula

The quadratic formula helps solve quadratic equations, which are polynomial equations of the form \(ax^2 + bx + c = 0\).
In the problem about profit, the quadratic equation was arranged as \(x^2 - 300x + 12500 = 0\).The quadratic formula itself is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
Here, \(a\), \(b\), and \(c\) are coefficients from the quadratic equation.
By using the quadratic formula, we can find the solutions or roots of the equation.
It is useful since not all quadratic equations can be easily factored, especially when they involve larger or more complex numbers.

Discriminant

The discriminant is a component of the quadratic formula and is given by \(b^2 - 4ac\).
It helps determine the number and type of solutions a quadratic equation has.

• If the discriminant is positive, the quadratic equation has two real and distinct solutions.
• If the discriminant is zero, the quadratic equation has exactly one real solution, often referred to as a `double root`.
• If the discriminant is negative, the quadratic equation has no real solutions but two complex solutions.

In the provided exercise, we found the discriminant to be 40000, which is positive.
This indicates that there are two real and distinct solutions for the number of ovens produced to meet the profit target.

Roots of Quadratic Equation

The roots of a quadratic equation are the values of \(x\) that satisfy the equation when it is set to zero.
These are the 'solutions' we find using methods like factoring, completing the square, or in this case, the quadratic formula.For the equation \(x^2 - 300x + 12500 = 0\), the quadratic formula gave us two potential solutions: \(x = 50\) and \(x = 250\).
We must consider logical constraints, such as the range given by the problem (\(0 \leq x \leq 200\)), to choose the valid root.
In this scenario, \(x = 50\) is the valid root.
Thus, a production of 50 ovens yields a profit of \$1250, while 250 ovens would have exceeded the allowed production range.