Problem 88 - Applied Calculus

Question

An electronics company's research budget is \(R(p)=3 p^{0.25},\) where \(p\) is the company's profit, and the profit is predicted to be \(p(t)=55+4 t,\) where \(t\) is the number of years from now. (Both \(R\) and \(p\) are in millions of dollars.) Express the research expenditure \(R\) as a function of \(t,\) and evaluate the function at \(t=5\).

Step-by-Step Solution

Answer

The research expenditure at \( t = 5 \) is approximately $10.03 million.

Step 1: Understand the Problem

We are given two functions: the research budget function \( R(p) = 3p^{0.25} \) which depends on profit, \( p \), and the profit function \( p(t) = 55 + 4t \) which depends on time, \( t \). Our task is to express \( R \) as a function of \( t \) and then evaluate this function at \( t = 5 \).

Step 2: Express Profit in Terms of Years

We substitute \( p(t) = 55 + 4t \) into \( R(p) \). This gives us the new function:\[ R(t) = 3(55 + 4t)^{0.25} \]

Step 3: Evaluate the Function at t = 5

To find the research budget in 5 years, substitute \( t = 5 \) into the function \( R(t) \):\[ R(5) = 3(55 + 4(5))^{0.25} \]First, calculate the profit in 5 years:\( p(5) = 55 + 4 \times 5 = 75 \).Substitute back into \( R(t) \):\( R(5) = 3(75^{0.25}) \).

Step 4: Calculate the Exponentiation and Final Result

Now calculate \( 75^{0.25} \). Using a calculator, \( 75^{0.25} \approx 3.3437 \). Then multiply this result by 3 to find the research expenditure:\( R(5) = 3 \times 3.3437 \approx 10.0311 \).

Key Concepts

Research Budget Function

The research budget function is a mathematical expression used to determine how much money is allocated to a company's research endeavors based on its profits. In the given scenario, the function is presented as:

• \( R(p) = 3p^{0.25} \)

This implies that the research budget, \( R \), grows proportionally with the profits, \( p \), raised to the power of 0.25. This type of function highlights the relationship between the company’s revenue and its investment in research, with more profits leading to higher research budgets.

To further understand, raise the profit value to the power of 0.25 means taking its fourth root, which implies gradual growth. Since the base of the exponent, 0.25, is less than 1, the growth of the research budget increases slowly as the profit grows. This mathematical approach can help businesses manage costs effectively and prioritize their spending on innovation based on actual available funds.

Profit Function

The profit function illustrates how profit increases over time within a specific model. It's given by:

• \( p(t) = 55 + 4t \)

Here, \( p \) represents profit in millions of dollars, while \( t \) marks the passage of time in years from the present. This linear equation implies that the company’s profit starts at \(55 million and increases by \)4 million each year. It models a straightforward and consistent profit growth.

When profit functions like this are used, they allow businesses to predict future earnings and plan budgets accordingly. For instance, by setting \( t = 5 \), one can calculate the expected profit at that time as \( 75 \) million dollars, encouraging informed financial decisions.

Time-Dependent Functions

Time-dependent functions are essential to understand how different variables interact and evolve over a time period. They create connections between time variables and other factors. In this exercise, linking the research budget to time through profit shows practical implications.

Modify a function such as the research budget (\( R(t) \)) by substituting the profit expressed in terms of time (\( p(t) \)) demonstrates this intertwined dependency:

• \( R(t) = 3(55 + 4t)^{0.25} \)

Such functions help a business anticipate resource allocation. For example, calculating \( R(t) \) gives insight into future spending needs. At \( t = 5 \), the expenditure becomes about $10.03 million, offering a foresight into financial requirements. This evaluative process enhances planning and acknowledges how various factors like profit and time correlate in real-world scenarios.