Question

A manufacturer of lawn chairs models the weekly production of chairs since 2009 by the function \(C(t)=100+35 t,\) where \(t\) is the time, in years, since 2009 and \(C\) is the number of chairs. The size of the workforce at the manufacturer's site is modelled by \(W(C)=3 \sqrt{C}\). a) Write the size of the workforce as a function of time. b) State the domain and range of the new function in this context.

Short answer

\(W(t) = 3\sqrt{100 + 35t}\) with domain \([0, \infty)\) and range \([30, \infty)\).

Step-by-step solution

Identify the given functions

The problem gives two functions: the production of chairs as a function of time, \(C(t) = 100 + 35t\), and the size of the workforce as a function of the number of chairs, \(W(C) = 3\sqrt{C}\).

Express the size of the workforce as a function of time

To find the size of the workforce as a function of time, first, substitute \(C(t)\) into \(W(C)\). If \(C(t) = 100 + 35t\), then \(W(t) = 3\sqrt{100 + 35t}\).

Write the new function

Therefore, the size of the workforce as a function of time is \(W(t) = 3\sqrt{100 + 35t}\).

Determine the domain of the new function

The domain is determined by the requirement that \(C\) (number of chairs) must be non-negative. So, \(100 + 35t \geq 0\). Therefore, \(t \geq 0\) since \(t\) is time in years since 2009.

Determine the range of the new function

Since \(C(t)\) can take any positive value starting from 100 and increasing as \(t\) increases, \(W(C) = 3\sqrt{C}\) will start from \(3\sqrt{100} = 30\) and increase without bound. Thus, the range is \([30, \infty)\).

Key concepts

domain and range

In mathematics, every function has a domain and a range. The domain of a function is the complete set of possible input values (usually represented by 'x') that the function can accept. Conversely, the range is the set of possible output values (usually represented by 'y') that the function can produce.

For the exercise given, we had two functions: \(C(t) = 100 + 35t\) and \(W(C) = 3\sqrt{C}\). Let's break down their domains and ranges:

• The function \(C(t) = 100 + 35t\) models the number of chairs produced over time. Here, 't' (time in years since 2009) starts at 0 and can increase indefinitely. Thus, the domain is \(t \geq 0\).
• For \(W(C) = 3\sqrt{C}\), the input 'C' represents the number of chairs, which must be non-negative. Therefore, the domain is \(C \geq 100\).

After composing these functions to get \(W(t)\), the domain remains \(t \geq 0\) since it is still measured in years from 2009. For the range of \(W(t)\), since it starts at \(3\sqrt{100} = 30\) and increases without bound, the range will be \([30, \infty)\).

function composition

Function composition involves creating a new function by applying one function to the result of another. It’s like a chain reaction: first, you apply one function to get a result, and then you plug that result into another function. The notation for function composition is \((f \circ g)(x)\), meaning you apply g first and then f.

For the given problem, we had two functions:

• Chairs production: \(C(t) = 100 + 35t\)
• Size of workforce: \(W(C) = 3\sqrt{C}\)

To express the size of the workforce as a function of time, we did the following:

• First, find the number of chairs as a function of time, \(C(t)\)
• Then, plug \(C(t)\) into \(W(C)\) to form \(W(t) = 3\sqrt{100 + 35t}\)

This composition simplifies complex relationships and helps us understand interconnected systems more effectively.

real-world applications of functions

Understanding and applying functions is crucial in real-world scenarios. They help us model and predict behavior in various fields, such as manufacturing, finance, biology, and technology.

In the provided exercise:

• \(C(t) = 100 + 35t\) helped model the production of lawn chairs over time, representing a practical relationship between time and production output.
• \(W(C) = 3\sqrt{C}\) linked the number of chairs produced to the size of the workforce, showcasing how personnel requirements change based on production levels.

This interplay of functions illustrates how companies can forecast the workforce needed to meet production goals. It’s a great example of function composition to solve real problems.

Other examples could be:

• In finance, predicting future investment returns based on current trends.
• In biology, modeling population growth in an ecosystem.
• In technology, estimating server loads based on user activity.

Functions and their compositions are powerful tools that help us model, analyze, and optimize various processes in our daily lives and work.