Question

1. mp modeling real life the equation y = 2x + 3 represents the cost y (in dollars) of mailing a package that weighs x pounds.

a. use a graph to estimate how much it costs to mail the package.
b. use the equation to find exactly how much it costs to mail the package.

Explanation:

Part (a)

Step1: Identify the weight

From the scale, the package weighs \( x = 1.26 \) pounds.

Step2: Analyze the cost equation

The cost equation is \( y = 2x + 3 \). To estimate using a graph (conceptually, we can think of plotting the line \( y = 2x + 3 \) and finding the \( y \)-value when \( x = 1.26 \)). First, find two points on the line: when \( x = 0 \), \( y = 3 \); when \( x = 2 \), \( y = 2(2)+3 = 7 \). Plot these points and draw the line. Then, find the \( y \)-value corresponding to \( x = 1.26 \). By linear approximation, since the line has a slope of 2, for \( x = 1 \), \( y = 2(1)+3 = 5 \); for \( x = 1.5 \), \( y = 2(1.5)+3 = 6 \). \( 1.26 \) is between 1 and 1.5. The difference between \( x = 1.26 \) and \( x = 1 \) is \( 0.26 \), so the increase in \( y \) is \( 2\times0.26 = 0.52 \). So \( y\approx5 + 0.52 = 5.52 \) dollars (this is an estimate from the graph).

Part (b)

Step1: Substitute the weight into the equation

The weight of the package \( x = 1.26 \) pounds, and the cost equation is \( y = 2x + 3 \). Substitute \( x = 1.26 \) into the equation.
\[
y = 2\times1.26 + 3
\]

Step2: Calculate the value

First, calculate \( 2\times1.26 = 2.52 \). Then, add 3: \( 2.52+3 = 5.52 \).

Answer:

a. The estimated cost from the graph is approximately \(\$5.52\) (the process involves graphing \( y = 2x + 3 \) and finding the \( y \)-value at \( x = 1.26 \)).
b. Using the equation, the cost is \(\$5.52\).