Problem 92 - Brief Applied Calculus

Question

Murrell's Rest Allowance Work-rest cycles for workers performing tasks that expend more than 5 kilocalories per minute \((\mathrm{kcal} / \mathrm{min})\) are often based on Murrell's formula $$ R(w)=\frac{w-5}{w-1.5} \quad \text { for } \quad w \geq 5 $$ for the number of minutes \(R(w)\) of rest for each minute of work expending \(w \mathrm{kcal} / \mathrm{min}\). Show that \(R^{\prime}(w)>0\) for \(w \geq 5\) and interpret this fact as a statement about the additional amount of rest required for more strenuous tasks.

Step-by-Step Solution

Answer

The derivative \( R'(w) > 0 \) shows that more strenuous tasks require more rest.

Step 1: Differentiate the Formula

We are given the formula \( R(w) = \frac{w-5}{w-1.5} \). To find \( R'(w) \), we need to apply the quotient rule for differentiation. The quotient rule is: if \( f(w) = \frac{u(w)}{v(w)} \), then \( f'(w) = \frac{u'(w)v(w) - u(w)v'(w)}{(v(w))^2} \). Here, \( u(w) = w-5 \) and \( v(w) = w-1.5 \).

Step 2: Compute Derivatives of Numerator and Denominator

Calculate the derivatives: \( u'(w) = \frac{d}{dw}(w-5) = 1 \) and \( v'(w) = \frac{d}{dw}(w-1.5) = 1 \).

Step 3: Apply the Quotient Rule

Now substitute back into the quotient rule: \[ R'(w) = \frac{(w-1.5) \times 1 - (w-5) \times 1}{(w-1.5)^2} \]. Simplify the numerator: \[ (w-1.5) - (w-5) = 5 - 1.5 = 3.5 \].

Step 4: Simplify the Derivative

Substitute the simplified numerator back: \[ R'(w) = \frac{3.5}{(w-1.5)^2} \]. Since \( (w-1.5)^2 > 0 \) for \( w \geq 5 \), \( R'(w) \) is positive.

Step 5: Interpretation of R'(w) > 0

Since \( R'(w) > 0 \), this means the function \( R(w) \) is increasing for \( w \geq 5 \). In practical terms, as the amount of energy expended \( w \) increases, the amount of rest \( R(w) \) required for each minute of work also increases. Thus, more strenuous tasks require more rest.

Key Concepts

Murrell's Formula

Murrell's formula is an essential concept used to determine the appropriate rest allowance for workers performing physically demanding tasks. This formula is particularly relevant when tasks involve expending more than 5 kilocalories per minute (kcal/min). It expresses the rest needed as a function of calories expended. The formula is represented as:\[ R(w) = \frac{w-5}{w-1.5} \]where \( R(w) \) is the rest in minutes, and \( w \) is the energy expenditure in kcal/min. The purpose of this formula is to provide a fair calculation of rest for varying levels of task difficulty. For instance, a higher caloric output necessitates more rest per minute of work, ensuring that workers can sustain their energy and health.
By calculating the derivative, we determine how changes in work intensity affect rest requirements. When \( R'(w) > 0 \), it signifies that as the task becomes more demanding, the need for rest increases.

Quotient Rule

The quotient rule is a fundamental technique in calculus differentiation used to find the derivative of a quotient of two functions. It is required when dealing with functions expressed as one function divided by another, written as \( \frac{u(w)}{v(w)} \). The rule states:\[ f'(w) = \frac{u'(w)v(w) - u(w)v'(w)}{(v(w))^2} \]In applying the quotient rule to Murrell's formula, we identify:

• \( u(w) = w-5 \),
• \( v(w) = w-1.5 \),
• \( u'(w) = 1 \),
• \( v'(w) = 1 \)

We substitute these into the quotient rule to find the derivative \( R'(w) = \frac{3.5}{(w-1.5)^2} \). This tells us how quickly the rest requirement increases as energy expenditure increases. The derivative being positive across all valid \( w \) values indicates an increasing function.

Work-Rest Cycles

Work-rest cycles are a crucial aspect of managing physical labor efficiently and safely. These cycles aim to balance the time spent working with appropriate rest periods, which is vital to prevent overexertion, reduce fatigue, and maintain productivity. The concept links closely with Murrell's formula, which calculates the ideal rest time based on caloric expenditure. As tasks become more strenuous, more rest is needed to allow workers to recover adequately.By implementing proper work-rest cycles, organizations can ensure their workers remain healthy and efficient. It is particularly important for tasks requiring high energy output, where physical strain is considerably greater. Understanding these cycles through the lens of calculus differentiation, as illustrated by changes in \( R(w) \), helps in designing work schedules that optimize human performance while minimizing risks associated with overwork.