https://r2.erweima.ai/mathbot/image/1726949863391orgrar2q.jpg

The function representing the productivity level \( A(t) \) of the workers is given by:

\[
A(t) = \frac{544}{4 + (t - 4.5)^2} + 28
\]

where \( t \) represents time in hours and \( t = 0 \) corresponds to 7:00 AM.

To determine at what time of the day the productivity level is maximized, we need to find the maximum of the function \( A(t) \). This involves taking the derivative of \( A(t) \) with respect to \( t \), setting it equal to zero, and solving for \( t \). Let's go through these steps.

### Step 1: Differentiate \( A(t) \)
We need to apply the quotient rule to differentiate the first term, and then set the derivative equal to zero to find the critical points.

Let me calculate that for you.The derivative of \( A(t) \) equals zero at \( t = 4.5 \). Since the problem states that \( t = 0 \) corresponds to 7:00 AM, we can add 4.5 hours to this time to find when the productivity is maximized.

### Step 2: Calculate the time
\[
7:00 \, \text{AM} + 4.5 \, \text{hours} = 11:30 \, \text{AM}
\]

Thus, the productivity level is maximized at 11:30 AM.

Would you like further details on the derivative calculation or the reasoning? Here are some related questions:

1. What is the quotient rule for differentiation?
2. How can we verify that \( t = 4.5 \) corresponds to a maximum?
3. What are critical points and how do we find them?
4. What does the graph of \( A(t) \) look like?
5. Can you explain the significance of the constants in the equation?

**Tip:** To verify if a critical point is a maximum, consider using the second derivative test.

El nivel de productividad de los operarios de una empresa se ha podido determinar mediante la función: A(t) = 544 / [4 + (t - 4.5)^2] + 28. Determine a qué hora del día el nivel de producción es máximo, con t = 0 correspondiente a las 7 A.M.

Find the time of day when worker productivity reaches its maximum using calculus. This problem involves differentiating the function A(t) = 544 / [4 + (t - 4.5)^2] + 28 and applying critical points analysis. The result shows that maximum productivity occurs at 11:30 AM.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation