Suppose that a shipping company’s postage rate for letters weighing 3.5oz or less is 50 cents for the first ounce a piecewise-defined function of the weight x of a letter, with 0. (Express your answer in dollars. )
The Correct Answer and Explanation is:
The problem refers to a postage rate function for a shipping company where the rate depends on the weight of the letter. The postage rate for letters weighing 3.5 ounces or less is 50 cents for the first ounce. From the way the question is framed, we are being asked to express this as a piecewise-defined function in terms of weight xxx.
Let’s define the function P(x)P(x)P(x) for the postage rate as a function of the weight of the letter:
- For letters weighing 1 ounce or less: The cost is 50 cents, since it’s the postage for the first ounce.
- For letters weighing between 1 and 3.5 ounces: The cost will increase, but we are told only the cost of the first ounce is 50 cents, with a linear increase for each additional ounce or fraction of an ounce.
Assume that each additional ounce or part of an ounce is charged at a fixed rate, such as 20 cents per additional ounce. This assumption is based on common postage rates, but can be adjusted if further information is provided.
Piecewise Function Definition:
The cost of postage can be defined as follows:
- If 0<x≤10 < x \leq 10<x≤1 ounces: The postage is P(x)=0.50P(x) = 0.50P(x)=0.50 dollars.
- If 1<x≤3.51 < x \leq 3.51<x≤3.5 ounces: The postage is P(x)=0.50+0.20(x−1)P(x) = 0.50 + 0.20(x – 1)P(x)=0.50+0.20(x−1) dollars. This accounts for the 50 cents for the first ounce and an additional 20 cents per ounce for each ounce greater than 1 ounce.
Explanation:
- The function is piecewise because the price changes depending on the weight of the letter.
- For letters weighing up to 1 ounce, the cost is fixed at 50 cents.
- For weights over 1 ounce but under or equal to 3.5 ounces, the cost increases by 20 cents for each additional ounce, calculated using the formula P(x)=0.50+0.20(x−1)P(x) = 0.50 + 0.20(x – 1)P(x)=0.50+0.20(x−1).
Thus, the postage rate for letters weighing 3.5 ounces or less is captured by this piecewise function. If you want the specific postage rate for a weight like 2 ounces or 3 ounces, you can substitute the value of xxx into the appropriate part of the function.
