Problem 51 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Mobile-phone ad spending between \(2005(t=1)\) and \(2011(t=7)\) is projected to be $$ S(t)=0.86 t^{0.96} \quad(1 \leq t \leq 7) $$ where \(S(t)\) is measured in billions of dollars and \(t\) is measured in years. What is the projected average spending per year on mobile-phone spending between 2005 and 2011 ?

Step-by-Step Solution

Answer

The projected average spending per year on mobile-phone ad spending between 2005 and 2011 is approximately \(\frac{0.86 \left[ \frac{7^{1.96}}{1.96} - \frac{1^{1.96}}{1.96} \right]}{7}\) billion dollars.

Step 1: Calculate the total mobile-phone ad spending

To find the total spending, we need to take the integral of the spending function \(S(t)\) in the interval \(1 \leq t \leq 7\). We have: $$ \int_1^7 S(t) dt = \int_1^7 0.86t^{0.96} dt $$ Now, we'll perform the integration.

Step 2: Integrate the function

To integrate the function, we'll use the power rule of integration: \(\int t^n dt = \frac{t^{n+1}}{n+1} + C\). $$ \int_1^7 0.86t^{0.96} dt = 0.86\int_1^7 t^{0.96} dt = 0.86 \left[ \frac{t^{0.96+1}}{0.96+1} \right]_1^7 $$

Step 3: Evaluate the definite integral

Now we'll evaluate the definite integral: $$ 0.86 \left[ \frac{t^{1.96}}{1.96} \right]_1^7 = 0.86 \left[ \frac{7^{1.96}}{1.96} - \frac{1^{1.96}}{1.96} \right] $$ This value represents the total mobile-phone ad spending between 2005 and 2011.

Step 4: Calculate the average spending per year

Now, we'll divide the total spending by the number of years in the interval (7 years) to find the average spending per year: $$ \text{Average spending per year} = \frac{0.86 \left[ \frac{7^{1.96}}{1.96} - \frac{1^{1.96}}{1.96} \right]}{7} $$ Calculating this value will give us the projected average spending per year on mobile-phone ad spending between 2005 and 2011.$$

Key Concepts

Integration

Integration is a fundamental concept in calculus, which allows us to find the accumulation of quantities over a certain interval. In essence, integration is like adding small pieces together to find the total or whole. When we integrate a function, we look for the antiderivative or the function whose derivative gives us the original function. This process is the reverse of differentiation.

There are several techniques of integration, but in this case, we use the power rule. The power rule for integration states that for any function of the form \( t^n \), the integral is given by \( \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \), where \( C \) is the constant of integration. This rule is applicable when \( n eq -1 \).

• To solve the problem, we first set up the definite integral which considers the upper and lower bounds of the interval. This accounts for the net area under the curve between specified points.
• We then evaluate this definite integral by finding the antiderivative and subtracting the values obtained at the upper and lower limits.

Once the function is integrated, the result reveals the total spending over the given period, which we use to calculate averages later on.

Average value of a function

The average value of a function over a certain interval gives a sense of the mean output the function achieves between two points. It's like finding the average of a set of numbers, but instead, we average out all possible outputs of a continuous function over a range.

To find the average value of a continuous function \( f(x) \) over the interval \([a, b]\), the formula is:

• \( \text{Average value} = \frac{1}{b-a} \int_a^b f(x) \, dx \)

Essentially, you integrate the function over the desired range and divide by the length of the interval. This process effectively compresses all accumulated values generated by the function into a single representative value.

• In the case of mobile-phone ad spending, we computed the definite integral over the 7-year span and divided it by 7, obtaining the average annual spending.
• This calculation helps in understanding what the spending would look like if it were evenly spread across all the years.

Understanding the average can be particularly valuable in business scenarios for budgeting and projection purposes.

Applied mathematics

Applied mathematics uses mathematical methods and formulas to solve real-world problems. It effectively connects theory with practice, making it indispensable in fields like engineering, economics, and finance.

In this exercise, applied mathematics helps in determining future ad spending, an important economic forecast. By using calculus, we can not only model the spending as a function but also analyze it to get useful insights over time.

• Integration, a tool from calculus, isn't just abstract math; it's applied to calculate something tangible—total ad spending in billions of dollars.
• The concept of average value gives a more understandable figure for stakeholders to analyze, enriching decision-making processes that rely on understanding trends and patterns over time.

Such applications highlight the significance of math outside the classroom, enabling accurate forecasting and informed decision-making based on numerical data. It also illustrates how mathematics supports and enhances strategic planning in technology and business sectors.