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Chapter 5: Problem 56

The cost, \(C(w),\) of mailing a large envelope weighing \(w\) ounces, \(0

Short Answer

Expert verified
Answer: The constants 0.88 and 1 in the cost function have the following interpretations: 1) 0.88 represents the base cost for mailing an envelope up to 1 ounce. 2) 1 is used to adjust the cost for any envelope weighing more than 1 ounce. The coefficient 0.17 represents the cost for every additional ounce above the base weight of 1 ounce. To mail a 9.1-ounce envelope, it would cost $2.41.

Step by step solution

01

Interpret constants 0.88 and 1

Both constants 0.88 and 1 are present in the equation of the cost function, \(C(w)=0.88+0.17(w-1)\). From this equation: 1) The constant 0.88 represents the base cost for mailing an envelope up to the weight of 1 ounce. That is, every envelope would cost at least 0.88 dollars. 2) The constant 1 is present in the formula to adjust any envelope weighing more than 1 ounce. If the envelope weighs less than or equal to 1 ounce, then (w-1) evaluates to 0(zero), and hence there will be no additional cost except the base cost ($0.88). #b)
02

Interpret the coefficient 0.17

The 0.17 is the cost for every additional ounce above the base weight of 1 ounce. So, for every ounce heavier than 1 ounce, the cost will increase by $0.17. #c)
03

Calculate the cost for 9.1-ounce envelope

The cost function given, \(C(w)=0.88 + 0.17(w-1)\), allows us to find the cost of mailing a large envelope weighing 9.1 ounces. The first step is to round up the weight to the nearest integer: \(w = \lceil 9.1 \rceil = 10\) Now, after substituting the value of \(w = 10\) into the equation: \(C(10) = 0.88 + 0.17(10-1) = 0.88 + 0.17(9) = 0.88 + 1.53 = 2.41\) Thus, to mail a 9.1-ounce large envelope, it would cost $2.41.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equation
A linear equation is a mathematical expression that can be used to describe relationships between two variables. In the context of the cost function, the linear equation is represented by \( C(w) = 0.88 + 0.17(w-1) \). Here, \( C(w) \) gives us the cost of mailing a large envelope based on its weight \( w \). A linear equation is made up of a constant term and a variable term, with each part having specific roles.
  • The constant \( 0.88 \) represents the base cost of mailing an envelope. This means that even if the envelope weighs just one ounce or less, you still have to pay this amount as a starting fee for the mailing process.
  • The term \( 0.17(w-1) \) accounts for any additional charges if the envelope weighs more than 1 ounce. Here \( 0.17 \) is a coefficient that scales with \( w-1 \).
Linear equations like this help us quickly calculate costs based on certain variables by plugging in different values and performing straightforward arithmetic.
Mathematical Modeling
Mathematical modeling is a systematic way of representing real-life problems through mathematical expressions and equations. This allows us to make predictions or understanding complex concepts by simplifying them into models. In our envelope mailing example, we use the linear equation \( C(w) = 0.88 + 0.17(w-1) \) to model the cost of sending envelopes.
Through mathematical modeling, we:
  • Define variables: \( w \) for weight and \( C(w) \) for cost.
  • Use parameters and constants, like \( 0.88 \) and \( 0.17 \), to represent fixed values in the real-world scenario.
  • Apply rounding, as in the weight \( w = \lceil 9.1 \rceil = 10 \), to align with practical considerations such as rounding up to the nearest ounce.
Mathematical modeling helps in creating a tool through which one can easily calculate outcomes, like determining mailing costs based on the weight of an envelope. This provides efficiency and clarity for decision making in logistics and pricing.
Weight Calculation
Weight calculation refers to determining the effective weight of an item to apply the correct cost or price. In the case of mailing a large envelope, the weight impacts the total mailing cost. With the given function \( C(w) = 0.88 + 0.17(w-1) \), weight is an essential variable.
We need to consider:
  • The range of weight \( 0 < w \leq 13 \), which means envelopes can weigh between just over 0 to 13 ounces.
  • Using rounding to the nearest upper whole number: if \( w = 9.1 \) ounces, it rounds up to \( w = 10 \) ounces. This rounding is important as it impacts the added costs over the 1-ounce base cost.
Understanding weight calculation is important, as it ensures accurate cost assessments when mailing or shipping items. This process ensures that any fractional value doesn鈥檛 negatively impact total costs, providing fairness and predictability in pricing.

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