Problem 10 - Algebra Form and Function

Question

The cost, \(\$ C\), of hiring a repairman for \(h\) hours is given by \(C=50+25 h\) (a) What does the repairman charge to walk in the door? (b) What is his hourly rate?

Step-by-Step Solution

Answer

Answer: The repairman's initial charge is $50, and the hourly rate is $25.

Step 1: Find the initial charge at h = 0

Plug the value \(h=0\) into the equation \(C=50+25h\) to find the initial charge. So, we have: \(C = 50 + 25(0) = 50\) For part (b), we need to determine the hourly rate from the equation. It's the coefficient of the \(h\) term, which represents how much the cost increases per hour.

Step 2: Read hourly rate from the equation

Look at the given equation: \(C = 50 + 25h\) The term "25h" tells us that the repairman charges 25 dollars for each hour of work. Therefore, the hourly rate is \(\$25\). So the repairman charges (a) \(\$50\) to walk in the door, and (b) has an hourly rate of \(\$25\).

Key Concepts

Cost Function

A cost function is a mathematical model that helps businesses and individuals understand how costs relate to various production activities. It translates real-world pricing information into a formula that can be analyzed and interpreted. In our exercise, the cost function is given by \[ C = 50 + 25h \] where \( C \) represents the total cost in dollars, and \( h \) signifies the number of hours worked by the repairman.

The function is composed of two main components:

• The constant term \( 50 \) represents the fixed charge or initial cost just for hiring the repairman. This cost applies regardless of how much work is done.
• The term \( 25h \) is the variable cost, depending on the hours worked. Each hour worked adds \$25 to the total cost.

Understanding how to interpret these components can make it easier to predict costs and make budget-friendly decisions.

Variable Coefficient

In a linear equation like our cost function, the variable coefficient is an essential element. This number directly influences how the variable term behaves within the function. In the function \[ C = 50 + 25h \]the variable coefficient is \( 25 \), found in the term \( 25h \).

Here's why the variable coefficient is important:

• It signifies the rate at which the cost increases for each additional hour worked. In this example, every hour adds $25 to the cost, as indicated by the coefficient \( 25 \).
• Understanding the coefficient helps you predict how changes in the variable (hours worked) will impact the total cost.

Grasping the concept of the variable coefficient helps with more than just financial math. It’s a foundational idea in many areas of algebra and calculus.

Initial Value Problem

An initial value problem refers to determining the starting point of a function, often where the variable is zero. This is critical when calculating one-time fees or base costs in functions such as our cost function. For the equation \[ C = 50 + 25h \]the initial value problem involves setting \( h \) to zero to find the base cost. Solving this yields \[ C = 50 + 25(0) = 50 \]
This tells us that the initial charge, or the fee just for the repairman to show up, is \$50.

Whether you're working on a math problem or hiring someone for services, spotting and calculating initial values can provide clarity on starting costs before considering any variable influences.