Problem 100 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

As a result of increasing energy costs, the growth rate of the profit of the 4-yr old Venice Glassblowing Company has begun to decline. Venice's management, after consulting with energy experts, decides to implement certain energy-conservation measures aimed at cutting energy bills. The general manager reports that, according to his calculations, the growth rate of Venice's profit should be on the increase again within 4 yr. If Venice's profit (in hundreds of dollars) \(t\) yr from now is given by the function $$ P(t)=t^{3}-9 t^{2}+40 t+50 \quad(0 \leq t \leq 8) $$ determine whether the general manager's forecast will be accurate. Hint: Find the inflection point of the function \(P\) and study the concavity of \(P\).

Step-by-Step Solution

Answer

The inflection point of the profit function \(P(t)\) occurs at \(t=3\). Before the inflection point, the function is concave down, indicating a decrease in profit growth rate. After the inflection point, the function is concave up, indicating an increase in profit growth rate. Therefore, the general manager's forecast is accurate, and the growth rate of Venice's profit will increase again within 4 years.

Step 1: Find the first and second derivative of the function

Let's find the first and second derivative of the given profit function \(P(t)\). The first derivative represents the rate of change of the profit function, i.e., the profit growth rate: $$ P'(t) = \frac{dP}{dt} = 3t^2 - 18t + 40 $$ Now, let's find the second derivative, which will help us analyze the concavity of the function: $$ P''(t) = \frac{d^2P}{dt^2} = 6t - 18 $$

Step 2: Find the inflection point

To find the inflection point, we need to find when the second derivative equals zero: $$ P''(t) = 6t - 18 = 0 $$ Solving for \(t\): $$ t = \frac{18}{6} = 3 $$ The inflection point occurs at \(t = 3\).

Step 3: Analyze the concavity of the function

To analyze the concavity of the function, we need to check the sign of the second derivative before and after the inflection point \(t = 3\). 1. Choose a value of \(t < 3\), say \(t = 2\): $$ P''(2) = 6(2) - 18 = -6 $$ Since \(P''(2) < 0\), the function is concave down (i.e., there's a decrease in the profit growth rate) before the inflection point. 2. Choose a value of \(t > 3\), say \(t = 4\): $$ P''(4) = 6(4) - 18 = 6 $$ Since \(P''(4) > 0\), the function is concave up (i.e., there's an increase in the profit growth rate) after the inflection point. Since the concavity of the function changes from negative to positive around the inflection point, we can conclude that the general manager's forecast is accurate. The growth rate of Venice's profit will indeed increase again within 4 years.

Key Concepts

Derivative

Derivatives are a fundamental concept in calculus that describe how a function changes at any given point. When we calculate the derivative of a function, we are finding the rate at which that function's value is changing. In the context of business, it can refer to how something like profit is growing or declining over time. For the function provided, which models the profit over time for the Venice Glassblowing Company, the first derivative, denoted as \( P'(t) \), represents the rate of change of profits. It tells us if the profit is increasing or decreasing at any particular point by taking the derivative of \( P(t) = t^3 - 9t^2 + 40t + 50 \). Using standard differentiation rules, the derivative is \( P'(t) = 3t^2 - 18t + 40 \). This result helps predict the trend of the company's profits over the years.

Inflection Point

An inflection point is a point on a curve where the concavity changes. In simpler terms, it is where the curve changes from being "smiley" (concave up) to "sad" (concave down) or vice versa. Inflection points provide valuable insights into the behavior of profit growth or decay. For the Venice Glassblowing Company's profit function, the inflection point is identified by setting the second derivative equal to zero. This is based on the second derivative \( P''(t) = 6t - 18 \). By solving \( 6t - 18 = 0 \), we find that the inflection point occurs at \( t = 3 \). This point suggests a tricky but crucial transition where the behavior of the profit changes significantly. By locating this change, business strategists can make more informed decisions about the timing of their actions.

Concavity Analysis

Concavity analysis lets us determine whether a function curve is bending upwards or downwards at any given interval. This analysis provides key insights, such as whether the rate of profit increase is itself increasing (concave up) or decreasing (concave down). For the profit function \( P(t) \), examining the sign of the second derivative \( P''(t) \) enlightens us about its concavity. A negative second derivative indicates a concave down shape, meaning growth is slowing, while a positive sign indicates concave up, meaning growth is accelerating. Analysts calculated \( P''(2) = -6 \) before the inflection point, showing it was concave down, suggesting profit growth decline. After the inflection point, with \( P''(4) = 6 \), the function is concave up, conveying that profits are likely on the rise. This shifts understanding and supports the prediction of increased profit rates from year 3 onwards.