Question

lesson 12: piecewise functions
cool down: international postage
p is a function that gives the cost, in dollars, of mailing a letter from the united states to mexico in 2018 based on the weight of the letter in ounces, w.
the function is defined by this set of rules: $p(w)=\begin{cases}1.15, & 0 < wleq1\\1.72, & 1 < wleq2\\2.29, & 2 < wleq3\\2.86, & 3 < wleq3.5end{cases}$

1. how much does it cost to send a letter that weighs 1.5 ounces? 2 ounces?
2. sketch a graph of the function on the coordinate plane.

Explanation:

Step1: Determine cost for 1.5 ounces

Since \(1 < 1.5\leq2\), use \(P(w)=1.72\) when \(1 < w\leq2\).

Step2: Determine cost for 2 ounces

Since \(1 < 2\leq2\), use \(P(w)=1.72\) when \(1 < w\leq2\).

Step3: Sketch - for \(0 < w\leq1\)

The function value is \(P(w) = 1.15\), so we have a horizontal line segment from \(w = 0\) (not included) to \(w = 1\) at \(y=1.15\).

Step4: Sketch - for \(1 < w\leq2\)

The function value is \(P(w)=1.72\), so we have a horizontal line segment from \(w = 1\) (not included) to \(w = 2\) at \(y = 1.72\).

Step5: Sketch - for \(2 < w\leq3\)

The function value is \(P(w)=2.29\), so we have a horizontal line segment from \(w = 2\) (not included) to \(w = 3\) at \(y=2.29\).

Step6: Sketch - for \(3 < w\leq3.5\)

The function value is \(P(w)=2.86\), so we have a horizontal line segment from \(w = 3\) (not included) to \(w = 3.5\) at \(y=2.86\).

Answer:

1. The cost of sending a 1.5 - ounce letter is \(1.72\) dollars. The cost of sending a 2 - ounce letter is \(1.72\) dollars.
2. The graph consists of horizontal line - segments: \(y = 1.15\) for \(0 < w\leq1\), \(y = 1.72\) for \(1 < w\leq2\), \(y = 2.29\) for \(2 < w\leq3\), and \(y = 2.86\) for \(3 < w\leq3.5\).