Problem 58 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The number of people watching TV on mobile phones is expected to grow at the rate of $$ N^{\prime}(t)=\frac{5.4145}{\sqrt{1+0.91 t}} \quad(0 \leq t \leq 4) $$ million/year. The number of people watching TV on mobile phones at the beginning of \(2007(t=0)\) was \(11.9\) million. a. Find an expression giving the number of people watching TV on mobile phones in year \(t .\) b. According to this projection, how many people will be watching TV on mobile phones at the beginning of \(2011 ?\)

Step-by-Step Solution

Answer

a. The expression giving the number of people watching TV on mobile phones in year \(t\) is: $$ N(t) = \frac{5.4145}{0.91} \cdot 2(1+0.91t)^{1/2} + 5.9026 $$ b. According to this projection, at the beginning of 2011, approximately 20.7721 million people will be watching TV on mobile phones.

Step 1: Integrate \(N'(t)\) with respect to \(t\)

To find the function \(N(t)\) for the number of people watching TV on mobile phones, we need to integrate the rate function \(N'(t)\) with respect to \(t\). So, we will evaluate the following integral: $$ N(t) = \int N'(t) dt = \int \frac{5.4145}{\sqrt{1+0.91 t}} dt $$ #Step 2: Substitution#

Step 2: Substitute \(u = 1 + 0.91t\)

Let's use the substitution \(u = 1+0.91t\). Then, the derivative of \(u\) with respect to \(t\) is: $$ \frac{du}{dt} = 0.91 \implies dt = \frac{du}{0.91} $$ Now, substituting \(u\) and \(dt\) into the integral, we have: $$ N(t) = \int \frac{5.4145}{\sqrt{u}} \cdot \frac{du}{0.91} $$ #Step 3: Evaluate the integral#

Step 3: Evaluate the integral

Now we can evaluate the integral: $$ N(t) = \frac{5.4145}{0.91} \int u^{-1/2} du = \frac{5.4145}{0.91} \cdot 2u^{1/2} + C $$ Substitute back \(u = 1 + 0.91t\): $$ N(t) = \frac{5.4145}{0.91} \cdot 2(1+0.91t)^{1/2} + C $$ #Step 4: Solve for C using the initial condition#

Step 4: Solve for C using the initial condition

At the beginning of 2007, the number of people watching TV on mobile phones is given as \(11.9\) million. Therefore, when \(t=0\), \(N(t) = 11.9\). Plugging this into our expression, we get: $$ 11.9 = \frac{5.4145}{0.91} \cdot 2(1+0.91(0))^{1/2} + C $$ Solving for \(C\), we find: $$ C = 11.9 - \frac{5.4145}{0.91} \cdot 2 = 5.9026 $$ Now we have the expression for the number of people watching TV on mobile phones in year \(t\): $$ N(t) = \frac{5.4145}{0.91} \cdot 2(1+0.91t)^{1/2} + 5.9026 $$ #Step 5: Find the number of people at the beginning of 2011#

Step 5: Evaluate \(N(t)\) at \(t=4\)

To find the number of people watching TV on mobile phones at the beginning of 2011, we need to evaluate \(N(t)\) at \(t=4\): $$ N(4) = \frac{5.4145}{0.91} \cdot 2(1+0.91(4))^{1/2} + 5.9026 \approx 20.7721 $$ At the beginning of 2011, approximately 20.7721 million people will be watching TV on mobile phones according to this projection.

Key Concepts

Integration

Integration is a fundamental concept in calculus and involves finding the antiderivative of a function. Essentially, integration is the reverse process of differentiation. In this exercise, the goal was to find the total number of people watching TV based on the growth rate, which is expressed as a derivative, \(N'(t)\). By integrating this rate, we determine the function \(N(t)\) that gives us the number of people at any time \(t\). To integrate \(N'(t) = \frac{5.4145}{\sqrt{1+0.91t}}\), we used substitution, a common technique to simplify the integral. We set \(u = 1 + 0.91t\), changing the variable to make integration straightforward. This method is powerful as it transforms complex problems into simpler forms. Always remember that the integral also includes a constant of integration \(C\), which we determine using initial conditions provided in the problem.

Differential Equations

Differential equations involve equations with derivatives, and they model how quantities change over time. In this exercise, \(N'(t)\) represents a differential equation, showing how the number of viewers grows. Solving differential equations typically involves finding a function, such as \(N(t)\), that satisfies the given rate of change.

• This involves moving from the derivative back to the original function, which is accomplished by integration.
• It's essential to take into account any initial conditions, like the initial number of people watching TV, to find unknowns such as the constant of integration \(C\).

Understanding differential equations helps us predict future trends and behaviors by modeling real-world phenomena, providing insight into how changes accumulate over time.

Mathematical Modeling

Mathematical modeling is the process of representing real-world scenarios with mathematical forms. This exercise exemplifies mathematical modeling by predicting how the number of mobile TV viewers will grow over time.

• The function \(N(t)\), derived from integrating the rate \(N'(t)\), is a model predicting future viewer numbers.
• Such models rely on assumptions and initial conditions. For instance, here the initial value given at \(t=0\), or 2007, allows us to accurately forecast this growth through 2011.
• Models like these are crucial not only in technology usage projections but also in fields like biology, economics, and physics.

By using these mathematical representations, one can simulate and estimate dynamics that are otherwise difficult to quantify directly, aiding in decision-making and planning.