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Chapter 12: Problem 54

The number \(N\) of employees and the total floor space \(S\) of your company are both changing with time. Show that the percentage rate of change of square footage per employee equals the percentage rate of change of \(S\) minus the percentage rate of change of \(N\). (The percentage rate of change of a quantity \(Q\) is \(\left.Q^{\prime}(t) / Q(t) .\right)\)

Short Answer

Expert verified
The percentage rate of change of square footage per employee (\( \frac{R^{\prime}(t)}{R(t)} \)) equals the percentage rate of change of total floor space (\( \frac{S^{\prime}(t)}{S(t)} \)) minus the percentage rate of change of the number of employees (\( \frac{N^{\prime}(t)}{N(t)} \)), as shown by the equation: \[ \frac{R^{\prime}(t)}{R(t)} = \frac{S'(t)}{S(t)} - \frac{N'(t)}{N(t)} \]

Step by step solution

01

Define the Given Variables and Their Rates of Change

We are given that the number of employees (N) and the total floor space (S) are time-dependent quantities. Let N(t) represent the number of employees and S(t) represent the total floor space at time t. We also know that the rates of change for both N and S are given by their derivatives with respect to time (t): N'(t) = rate of change of N with respect to time, S'(t) = rate of change of S with respect to time.
02

Calculate the Square Footage per Employee

Let ratio R(t) = S(t) / N(t), which represents the square footage per employee at any time t.
03

Differentiate R(t) with Respect to Time

To find the rate of change of square footage per employee, we need to differentiate R(t) with respect to time. We use the quotient rule to differentiate the quotient R(t) = S(t) / N(t): R'(t) = (S'(t)N(t) - S(t)N'(t)) / {N(t)}^2
04

Calculate the Percentage Rates of Change

We've been asked to express the result in terms of percentage rates of change. The percentage rate of change of S is \( \frac{S^{\prime}(t)}{S(t)} \), and the percentage rate of change of N is \( \frac{N^{\prime}(t)}{N(t)} \). Also, let the percentage rate of change of R be \( \frac{R^{\prime}(t)}{R(t)} \). Now, we want to find \( \frac{R^{\prime}(t)}{R(t)} \) in terms of the percentage rates of change of S and N.
05

Express R'(t) as the Difference of Percentage Rates of Change

We begin by dividing the derivative of R(t) (from Step 3) by R(t): \[ \frac{R^{\prime}(t)}{R(t)} = \frac{S'(t)N(t) - S(t)N'(t)}{S(t)N(t)^2 / N(t)} = \frac{S'(t)N(t) - S(t)N'(t)}{S(t)N(t)} \] \[ \frac{R^{\prime}(t)}{R(t)} = \frac{S'(t)}{S(t)} - \frac{N'(t)}{N(t)} \] Since \( \frac{R^{\prime}(t)}{R(t)} \) is the percentage rate of change of the square footage per employee, we now have: Percentage rate of change of R = Percentage rate of change of S - Percentage rate of change of N

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quotient Rule
To find the rate of change for a ratio like the square footage per employee, we use the Quotient Rule from calculus. This rule helps us differentiate a function that is the ratio of two other functions. The general formula for the Quotient Rule is given by:
\[ \frac{d}{dt}\left(\frac{u}{v}\right) = \frac{u'(t)v(t) - u(t)v'(t)}{(v(t))^2} \]
where \( u(t) \) and \( v(t) \) are functions of time, and \( u'(t) \) and \( v'(t) \) are their derivatives concerning time \( t \).
  • In our exercise, \( u(t) = S(t) \), representing the total floor space, and \( v(t) = N(t) \), representing the number of employees.
  • Therefore, the derivative \( R'(t) \) captures how the square footage per employee changes over time.
By applying this rule, we can accurately measure how the ratio changes, which is crucial for understanding percentage rates of change.
Derivative
A derivative represents the rate at which a function changes over time. For a function \( f(t) \), its derivative \( f'(t) \) gives us the slope or steepness at any point \( t \). Derivatives are central to finding how variables dynamically evolve, especially in contexts like business where employee numbers and office space might vary.
  • In our given case, the derivatives \( N'(t) \) and \( S'(t) \) tell us how the number of employees and the total floor space, respectively, change concerning time.
  • This allows businesses to track statistics like productivity or usage efficiency efficiently.
By understanding derivatives, we can predict future trends and make informed decisions based on how quickly or slowly things change over time.
Percentage Rate
The percentage rate of change is a way to compare how a quantity changes relative to its current state. It allows you to express this change as a percentage, making it easy to understand and communicate. In the context of our problem:
- The percentage rate of change of any quantity \( Q \) is given by \( \frac{Q'(t)}{Q(t)} \). This expression effectively shows you the relative change in \( Q \) per unit time.
  • For the total floor space \( S \), its percentage rate of change is \( \frac{S'(t)}{S(t)} \).
  • For the number of employees \( N \), it is \( \frac{N'(t)}{N(t)} \).
  • An essential insight from the initial exercise is that the percentage rate of change of square footage per employee \( R \) equals the difference between \( S \)'s and \( N \)'s percentage changes.
Understanding this concept is key because it allows companies to see clear growth patterns or declines, facilitating better strategic planning.

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Most popular questions from this chapter

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