Problem 78 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

According to a study conducted in 2004, the share of online advertisement, worldwide, as a percentage of the total ad market, is expected to grow at the rate of $$ R(t)=-0.033 t^{2}+0.3428 t+0.07 \quad(0 \leq t \leq 6) $$ percent/year at time \(t\) (in years), with \(t=0\) corresponding to the beginning of 2000 . The online ad market at the beginning of 2000 was \(2.9 \%\) of the total ad market. a. What is the projected online ad market share at any time \(t\) ? b. What was the projected online ad market share at the beginning of 2005 ?

Step-by-Step Solution

Answer

a. The projected online ad market share at any time \(t\) is given by the function \(S(t)\): $$ S(t) = -\frac{0.033}{3}t^3 + \frac{0.3428}{2}t^2 + 0.07t + 0.029 $$ b. The projected online ad market share at the beginning of 2005 is approximately 8.7%.

Step 1: Integrate R(t) to find the share function

To find the share function, we integrate R(t) with respect to t: $$ S(t) = \int [ -0.033t^2 + 0.3428t + 0.07] dt $$ Step 2: Calculate the indefinite integral

Step 2: Calculate the indefinite integral

Find the antiderivative of each term with respect to t: $$ S(t) = -\frac{0.033}{3}t^3 + \frac{0.3428}{2}t^2 + 0.07t + C $$ Where C is a constant of integration. Step 3: Determine the value of the constant C

Step 3: Determine the value of the constant C

At t=0, the ad market share is 2.9%. So we can use this information to find the value of C: $$ 0.029 = -\frac{0.033}{3}(0)^3 + \frac{0.3428}{2}(0)^2 + 0.07(0) + C $$ Therefore, C=0.029. Step 4: Write the share function S(t)

Step 4: Write the share function S(t)

Substitute the value of C back into the expression for S(t): $$ S(t) = -\frac{0.033}{3}t^3 + \frac{0.3428}{2}t^2 + 0.07t + 0.029 $$ a. The projected online ad market share at any time t is given by the function S(t): $$ S(t) = -\frac{0.033}{3}t^3 + \frac{0.3428}{2}t^2 + 0.07t + 0.029 $$ Step 5: Find the market share at the beginning of 2005

Step 5: Find the market share at the beginning of 2005

At the beginning of 2005, t=5 (since t=0 corresponds to the beginning of 2000). To find the market share at that time, evaluate the function S(t) at t = 5: $$ S(5) = -\frac{0.033}{3}(5)^3 + \frac{0.3428}{2}(5)^2 + 0.07(5) + 0.029 $$ Step 6: Calculate the projected online ad market share

Step 6: Calculate the projected online ad market share

Compute the numerical value for S(5): $$ S(5) \approx -\frac{0.033}{3}(125) + \frac{0.3428}{2}(25) + 0.07(5) + 0.029 \approx 0.087 $$ b. The projected online ad market share at the beginning of 2005 is 8.7% (approx).

Key Concepts

Integration

Integration is a mathematical process used to find areas under curves, among many other applications. In calculus, integration is the reverse process of differentiation, where we find the antiderivative or indefinite integral of a function. For the function given in the problem, \( R(t) = -0.033t^2 + 0.3428t + 0.07 \), integrating it with respect to \( t \) provides a new function \( S(t) \) that represents the accumulated growth. This is important because often we have a rate of change function, like \( R(t) \), and need to determine total accumulation over time.

When we integrate, we add a constant \( C \) to account for any initial conditions or starting points that aren't defined by the rate itself. In this context, \( C \) ensures that our solution corresponds to the known initial market share of online advertising.

Definite and Indefinite Integrals

Definite and indefinite integrals are key tools in calculus.
**Indefinite Integrals:**- These provide a general form of the antiderivative of a function.- They include a constant of integration \( C \), and answer the question: "What original function does this rate of change come from?"

For instance, finding \( S(t) \) by integrating \( R(t) \) is an indefinite integral:\[S(t) = -\frac{0.033}{3}t^3 + \frac{0.3428}{2}t^2 + 0.07t + C\]

**Definite Integrals:**- These evaluate the total change in a quantity between two points, \( a \) and \( b \), expressed as \( \int_a^b f(t) dt \).- In the context of this problem, calculating definite integrals would allow us to determine the change in market share over a specific interval, such as from \( t = 0 \) to \( t = 5 \).

By implementing both types of integrals appropriately, we can solve a wide range of practical problems involving change and accumulation in fields beyond just advertising.

Mathematical Modeling

Mathematical modeling is the process of designing a mathematical construct, like our function \( R(t) \), to represent and analyze real-world situations. It allows predictions about events or behaviors over time. In this exercise, \( R(t) \) models the growth rate of online advertisement share over time, based on data collected in 2004. The function combines concepts of calculus with real-world data, providing a simulated framework to understand potential advertising trends.

This model:

• Provides a rate \( R(t) \) that is polynomial, indicating complexity beyond simple exponential or linear growth.
• Offers a practical use for calculus, as differentiating or integrating can yield insights into both instantaneous rates and accumulated changes.

Understanding mathematical models like this one helps us traverse between theory and application, forecasting outcomes and aiding decision-making processes based on existing predictions.

Rate of Change

The concept of the rate of change is central to calculus and understanding how quantities evolve over time. In this example, the rate of change, \( R(t) \), indicates how the percentage of online ads relative to the total ad market adjusts annually.

Consider:

• This rate of change function incorporates terms like \(-0.033t^2\), which suggest the rate itself can speed up or slow down over time.
• By evaluating \( R(t) \), businesses can anticipate increases or decreases in their market strategies.

Calculating the rate of change is essential as it provides insights into the dynamics of the growth situation. It helps businesses adjust strategies for maximizing market share based on anticipated trends, highlighting the importance of calculus in strategic planning. By understanding how fast or slow a particular change occurs, we can make informed predictions and adaptations to meet future challenges effectively.