Question

The equation relating the number of customized cell phones produced and the profit per cell phone is \(p=-2.50 n^2+21 n\), where \(n\) is in 100,000s. Plot the resulting graph. Be sure to label and number the axes appropriately and indicate the maximum value of \(p\).

Short answer

The maximum profit is 44.1 when 420,000 cell phones are produced. Plot the graph with the x-axis (phones produced) and y-axis (profit).

Step-by-step solution

Identify the variables and parameters

The equation is given by \(p = -2.50 n^2 + 21 n\), where \(p\) is the profit per cell phone and \(n\) is the number of customized cell phones produced in units of 100,000.

Determine the vertex of the parabola

Rewrite the equation in the standard form of a quadratic equation \(p = an^2 + bn + c\). Here, \(a = -2.50\), \(b = 21\), and \(c = 0\). The maximum value of a downward-facing parabola is at \(n = -\frac{b}{2a}\). Calculate \(n = -\frac{21}{2 \times -2.50} = 4.2\).

Calculate the maximum profit

Substitute \(n = 4.2\) into the equation to find the maximum profit: \(p = -2.50(4.2)^2 + 21(4.2) = 44.1\). So the maximum profit is \(p = 44.1\).

Set up the graph

Plot the points by choosing different values of \(n\) (both less than and greater than 4.2) and computing corresponding \(p\) values using the equation \(p = -2.50 n^2 + 21 n\).

Draw the graph

Draw the parabola using the points obtained in Step 4. Ensure to label the x-axis with the number of phones produced (\(n\) in units of 100,000) and the y-axis with the profit per phone (\(p\)). Also, mark the maximum point \((4.2, 44.1)\) on the graph.

Indicate the maximum value

Clearly indicate on the graph that the maximum profit of \(p = 44.1\) occurs when \(n = 4.2\), which corresponds to 420,000 phones produced.

Key concepts

parabola

In mathematics, a parabola is a specific type of curve on a graph. It's the graph of a quadratic equation, usually shaped like a U or an inverted U. For the equation given in the exercise, which is a downward-facing parabola, the formula is \[p = -2.50 n^2 + 21 n\] Here, 'p' stands for profit per cell phone, and 'n' stands for the number of customized cell phones produced in units of 100,000. The minus sign before the term \(n^2\) indicates that the parabola opens downwards. This shape is important because it helps us understand the relationship between production numbers and profit. A parabola has certain features: a vertex, which is the highest or lowest point, an axis of symmetry, and can open upwards or downwards.

maximum profit

The concept of maximum profit is critical in scenarios like these. Since the parabola opens downwards, the vertex of the parabola represents the point of maximum profit. By finding this point, a business can determine the optimal number of cell phones to produce. In the given equation, the vertex gives us this information. The formula for the maximum value in a downward-facing parabola, given by \[p = an^2 + bn + c\], is found at \(n = -\frac{b}{2a}\) When we substitute the given values: \(a = -2.50\) and \(b = 21\), we find that \(n = 4.2\). This means the maximum profit occurs when 420,000 cell phones are produced. You can then calculate the maximum profit by substituting \(n = 4.2\) back into the original equation.

vertex calculation

Vertex calculation is essential to find the maximum or minimum values in quadratic equations. In our scenario, we use the formula for calculating the vertex of the parabola: \(n = -\frac{b}{2a}\). Substitute \(a = -2.50\) and \(b = 21\) into the formula. Solving this, \(n = -\frac{21}{2 \times -2.50}\). That simplifies to: \(n = 4.2\). Now, substitute \(n = 4.2\) back into the profit equation to find the maximum profit. The maximum profit \(p\) is calculated as: \(p = -2.50(4.2)^2 + 21(4.2)\). So, \(p = 44.1\). Thus, the maximum profit is 44.1 units, when 420,000 phones are produced.

graphing equations

Graphing equations, especially quadratics, helps visualize relationships between variables. Here, plotting \(p = -2.50 n^2 + 21 n\) gives a clear visual of how the number of produced phones affects profit. Follow these steps to graph:

• Label the x-axis as 'Number of Phones (in 100,000's)'.
• Label the y-axis as 'Profit per Phone'.
• Plot points for different values of \(n\), including the vertex point \( (4.2, 44.1)\).
• Draw the parabola connecting these points, ensuring it's downward-facing.
• Highlight the vertex point to indicate maximum profit.

This way, you can easily see the optimal number of units to produce for maximum profitability alongside how profits change with varying production levels.