Problem 62 - Algebra and Trigonometry

Question

Sales A soft-drink vendor at a popular beach analyzes his sales records, and finds that if he sells \(x\) cans of soda pop in one day, his profit (in dollars) is given by $$ P(x)=-0.001 x^{2}+3 x-1800 $$ What is his maximum profit per day, and how many cans must he sell for maximum profit?

Step-by-Step Solution

Answer

Maximum profit is $450 when selling 1500 cans.

Step 1: Understand the Problem

The given problem provides a profit function in the form of a quadratic equation: \(P(x) = -0.001x^2 + 3x - 1800\). This is a downward opening parabola because the coefficient of \(x^2\) is negative. We need to find the maximum profit and the corresponding number of soda cans sold to achieve this profit.

Step 2: Identify the Vertex of the Parabola

For a quadratic equation \(ax^2 + bx + c\), the vertex form gives the maximum or minimum value. The vertex \(x\)-coordinate for maximum profit is given by \(x = -\frac{b}{2a}\). Here, \(a = -0.001\) and \(b = 3\).

Step 3: Calculate the Vertex \(x\)-coordinate

Substitute \(a\) and \(b\) into the formula: \(x = -\frac{3}{2 \times -0.001} = \frac{3}{0.002} = 1500\). Thus, the vendor must sell 1500 cans for maximum profit.

Step 4: Calculate Maximum Profit

To find the maximum profit, substitute \(x = 1500\) into the profit equation: \(P(1500) = -0.001(1500)^2 + 3(1500) - 1800\).

Step 5: Simplify and Solve

Calculate \(P(1500)\): \(-0.001(1500)^2 = -2250\), \(3(1500) = 4500\). Thus, \(P(1500) = -2250 + 4500 - 1800 = 450\).

Step 6: Conclusion

The maximum profit per day is $450, achieved by selling 1500 cans of soda.

Key Concepts

Maximum Profit

Understanding maximum profit in a business scenario often involves mathematical concepts, especially with quadratic functions. In our example, the profit function is expressed as a quadratic equation: \[ P(x) = -0.001x^2 + 3x - 1800 \]. In this equation, profit is dependent on the number of cans sold, represented by \(x\). The goal is to find the point at which this profit is the highest. This peak is known as the maximum profit. For any profit function structured like a downward opening parabola, this peak will be at the vertex. Finding this point involves some straightforward calculations, which we'll detail in the next sections.

Vertex of a Parabola

The vertex of a parabola is a crucial concept when trying to determine maximum or minimum values in quadratic functions. For downward opening parabolas (where the \(x^2\) term's coefficient is negative), the vertex represents the maximum point. The quadratic formula for finding the \(x\)-coordinate of the vertex is: \[ x = -\frac{b}{2a} \] In our scenario, with the profit function given as \(P(x) = -0.001x^2 + 3x - 1800\), the coefficients \(a = -0.001\) and \(b = 3\). Plugging into the formula: \[ x = -\frac{3}{2 \times -0.001} = 1500 \]. Therefore, the vendor reaches maximum profit by selling 1500 cans. This vertex calculation is essential in many business applications, helping predict the most profitable outcome.

Profit Function

The profit function is a mathematical representation of how profit changes with different inputs. In this case, the profit function provides how the number of soda cans sold (\(x\)) impacts the overall profit. Our equation \(P(x) = -0.001x^2 + 3x - 1800\) shows:

• The negative \(x^2\) term indicates a parabola that opens downwards, signifying that there is a peak (maximum profit) and then profit declines as \(x\) increases beyond this point.
• The linear \(x\) term shows the rate at which profit changes with the number of cans sold.
• The constant term (-1800) gives the initial loss or expense before any cans are sold.

This function helps in understanding revenue trends and identifying optimized selling strategies by clearly marking when profits will start to decrease.

Parabolas

Parabolas are critical in various mathematical and real-world applications, particularly in analyzing quadratic functions. They are U-shaped curves described by an equation of the form \(ax^2 + bx + c\). Depending on the sign of \(a\):

• If \(a\) is positive, the parabola opens upwards, indicating a minimum point (bottom of the U).
• If \(a\) is negative, as in our problem \(a = -0.001\), the parabola opens downwards, and has a maximum point (top of the U).

For businesses, knowing the direction and vertex of a parabola can be pivotal in determining strategies to maximize profit, like finding the ideal number of product units to optimize revenue. Parabolas thus serve as a visual and analytical tool in decision-making processes.