Problem 88 - Brief Applied Calculus

Question

BUSINESS: Research Expenditures 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 function is \( R(t) = 3(55 + 4t)^{0.25} \) and \( R(5) \approx 7.971 \) million dollars.

Step 1: Understand the Given Functions

We are given two functions: the research budget function \( R(p) = 3p^{0.25} \) for the research expenditures, and the profit function \( p(t) = 55 + 4t \) that models the predicted profit over time (in years). Our goal is to express the research budget as a function of time \( t \).

Step 2: Substitute the Profit Function into the Research Budget Function

Since the research budget \( R \) is dependent on profit \( p \) and the profit is a function of time \( t \), we substitute \( p(t) = 55 + 4t \) into \( R(p) \). This gives us a new function: \( R(t) = 3(55 + 4t)^{0.25} \).

Step 3: Evaluate the Research Expenditure Function at t=5

Substitute \( t = 5 \) into the function \( R(t) \) to find the research expenditures at that point in time. Calculate: \[R(5) = 3(55 + 4 \times 5)^{0.25} = 3(55 + 20)^{0.25} = 3(75)^{0.25}\]

Step 4: Simplify and Calculate

Now compute \( 75^{0.25} \), which is the fourth root of 75. Use a calculator to find \( 75^{0.25} \approx 2.657 \), and then multiply by 3 for the final expenditure: \[R(5) = 3 \times 2.657 \approx 7.971\]Thus, the research expenditure at \( t = 5 \) years is approximately 7.971 million dollars.

Key Concepts

Functions and Modeling

Functions and modeling help us simplify and analyze complex real-world situations mathematically. In this exercise, we have two main functions: one representing a company's research expenditure and the other its profit over time.

The research budget function, given as \( R(p) = 3p^{0.25} \), tells us how much is allocated to research based on the profit \( p \). Profit itself is modeled by the function \( p(t) = 55 + 4t \). This shows how the profit changes over time based on the variable \( t \), representing years.

By substituting the profit function into the research budget function, we align the two models into one cohesive representation. This modeling technique is crucial in applied mathematics to predict future outcomes, like budget allocations based on changing profits. It simplifies the process of understanding how time affects expenditure directly.

Profit Functions

Profit functions are essential in understanding financial growth over time. Here, the profit function \( p(t) = 55 + 4t \) is a linear representation showing a steady increase in profit. Each year, the profit increases by 4 million dollars, starting from an initial profit of 55 million dollars.

By expressing profit this way, companies can predict their financial trajectory and make informed decisions about future investments. Understanding the linear growth pattern provides clarity about the expected stability and growth rate of profits.

In our scenario, the function allows us to see what the profit will be at any number of years \( t \). This insight is vital for strategic planning and ensuring that sufficient resources are allocated to fund research projects or other business activities.

Applied Mathematics

Applied mathematics bridges the gap between abstract mathematical theory and practical problems encountered in the real world. When tackling this exercise, we utilize principles of applied mathematics to interconnect the profit and research expenditure functions.

This approach helps us convert a theoretical problem into a practical equation that allows businesses to evaluate how budget will adjust with time. Specifically, substituting \( p(t) = 55 + 4t \) into \( R(p) = 3p^{0.25} \) molds a function over time, \( R(t) \), that directly reflects how research expenditures evolve.

Such mathematical applications empower decision-makers to compute future budgets conveniently, predict financial needs, and set fiscal priorities—essentially making complicated financial predictions approachable and manageable.