Problem 99 - Intermediate Algebra

Question

The rational function $$ f(t)=\frac{t^{2}+2 t}{2 t+2} $$ gives the number of days it would take two webpage designers, working together, to design a standard website for a business that designer 1 (working alone) could complete in \(t\) days and designer 2 (working alone) could complete in \(t+2\) days. a. If designer 1 could complete the website in 15 days, how long would it take both designers working together? b. If designer 2 could complete a website in 20 days, how long would it take both designers working together?

Step-by-Step Solution

Answer

a. Approximately 8 days. b. Approximately 9.47 days.

Step 1: Understanding the Function

The given function is \[ f(t) = \frac{t^2 + 2t}{2t + 2} \]This represents the time it takes for two designers working together to complete a job. Here, \(t\) is the time designer 1 takes to complete the job alone, and \(t + 2\) is the time for designer 2.

Step 2: Solving Part (a)

For part (a), designer 1 completes the job in \(t = 15\) days. Substitute \(t = 15\) into the function:\[ f(15) = \frac{15^2 + 2(15)}{2(15) + 2} = \frac{225 + 30}{30 + 2} = \frac{255}{32} = 7.96875 \]This means that both designers working together will take approximately 8 days.

Step 3: Solving Part (b)

For part (b), designer 2 takes 20 days, so \(t + 2 = 20\), which gives \(t = 18\). Substitute \(t = 18\) into the function:\[ f(18) = \frac{18^2 + 2(18)}{2(18) + 2} = \frac{324 + 36}{36 + 2} = \frac{360}{38} = 9.473684 \]Thus, both designers working together will take approximately 9.47 days.

Key Concepts

Work Problems

Work problems in algebra involve situations where multiple entities work together to accomplish a task. In this case, our task is to figure out how long it will take for two webpage designers to complete a website when working collaboratively. The core idea is to understand how individual work rates combine.

For example, if designer 1 can complete a website in 15 days, his work rate is \( \frac{1}{15} \) of the website per day. Similarly, if designer 2 requires 17 days to complete the same website alone (as given by the equation \( t + 2 \)), his rate is \( \frac{1}{17} \) per day.

When they work together, the rates are combined by adding these individual rates together. This can be expressed as:

• Combined Work Rate: \( \frac{1}{t} + \frac{1}{t+2} \)
• Total Days = \( \frac{1}{{ \frac{1}{t} + \frac{1}{t+2} }} \)

Understanding this concept of combining individual rates helps us solve a wide range of real-world problems involving teamwork and collaboration.

Collaborative Work

Collaborative work focuses on how multiple contributors can complete tasks more efficiently together than individually. In our problem, the rational function \( f(t) = \frac{t^2 + 2t}{2t + 2} \) encapsulates this notion through algebraic means.

This function shows that the combined time required for two designers is generally less than either working alone. Taking the case solved above, if designer 1 completes the task in 15 days and designer 2 in 17 days, their collaborative effort reduces the joint completion time to approximately 8 days.

Working collaboratively often leads to improved efficiency because:

• Tasks are divided and more expertise can be utilized.
• Unique skills contribute to greater overall productivity.
• The burden on individual contributors is reduced, which can lead to faster completion.

This concept is widely applied in various contexts, from team projects to assembly line work, and highlights the benefits of shared goals and teamwork in achieving a common objective.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. In rational functions like our example, expressions help in understanding relationships between quantities.

For instance, the function \( f(t) = \frac{t^2 + 2t}{2t + 2} \) is a rational expression. The numerator \( t^2 + 2t \) stems from the algebraic manipulation that calculates the overlapping contributions of two designers’ work. The denominator \( 2t + 2 \) represents combined output over a period.

In algebra, simplifying such expressions involves canceling out terms and solving for unknowns. This leads to clearer insights and solutions:

• The parts of an expression (coefficients, terms, and operators) communicate the rate and time interrelationships.
• The manipulations, such as factoring and expanding, enable solving equations efficiently.
• Recognizing patterns in expressions aids in quick problem-solving.

A firm grasp on algebraic expressions not only aids in solving academic problems but also empowers logical thinking for real-life situations and professional fields where precision and clarity are critical.