The cost of mailing a first class letter is $0.46 for the first ounce and $0.20 for each additional ounce or portion of an ounce. the letter y represents the total cost for mailing a letter that weighs x ounces. which type of function model would
The correct answer and explanation is :
The situation described, where the cost of mailing a letter is based on its weight in ounces, is best modeled by a piecewise function.
Explanation of the Function:
- First ounce cost: For the first ounce of a letter, the cost is a fixed $0.46. This means that for any letter weighing up to 1 ounce, the cost is always $0.46.
- Additional ounces: For each additional ounce, the cost increases by $0.20. If the letter weighs more than 1 ounce, the additional cost depends on how many full or partial ounces are above the first ounce. For example, if the letter weighs 2 ounces, the total cost is $0.46 for the first ounce plus $0.20 for the second ounce, making the total cost $0.66.
The Piecewise Function Model:
A piecewise function is one that has different expressions for different intervals of the independent variable. In this case, the independent variable is the weight of the letter in ounces (denoted by xx), and the dependent variable is the total cost of mailing the letter (denoted by yy).
The piecewise function can be written as follows: y(x)={0.46if 0<x≤10.46+0.20(x−1)if x>1y(x) = \begin{cases} 0.46 & \text{if } 0 < x \leq 1 \\ 0.46 + 0.20(x – 1) & \text{if } x > 1 \end{cases}
Why a Piecewise Function?
- For x ≤ 1 (the first ounce), the cost is simply $0.46, so the first part of the function gives the price for letters that are 1 ounce or less.
- For x > 1 (greater than 1 ounce), the cost increases by $0.20 for each additional ounce, so the second part of the function accounts for the additional cost of each ounce above 1. This part of the function adds $0.20 multiplied by the excess weight beyond the first ounce (i.e., x−1x – 1).
Practical Use:
This piecewise function helps calculate the total cost for any weight xx by evaluating which condition the weight falls under. For instance:
- For a 1-ounce letter, substitute x=1x = 1 into the first case to get y=0.46y = 0.46.
- For a 2-ounce letter, substitute x=2x = 2 into the second case to get y=0.46+0.20(2−1)=0.66y = 0.46 + 0.20(2 – 1) = 0.66.
In conclusion, a piecewise function model is the correct approach because the cost structure is determined by different rules for different weight categories, making it a natural fit for this scenario.