Question

the student council is ordering customized foam hands to promote school spirit. the table shows the cost of ordering different numbers of foam hands. the company charges so much for each foam hand ordered and a processing fee when ordering. how much do they charge per foam hand? (slope) how much is the processing fee? (y-intercept) number of foam hands: 4, 6, 8, 10, 12; cost (dollars): 34, 46, 58, 70, 82

Explanation:

Part 1: Cost per foam hand (Slope)

Step1: Choose two points

Let's take the points \((4, 34)\) and \((6, 46)\) from the table.

Step2: Use slope formula

The slope \(m\) (which represents the cost per foam hand) is calculated using the formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
Substituting \(x_1 = 4\), \(y_1 = 34\), \(x_2 = 6\), and \(y_2 = 46\) into the formula:
\(m=\frac{46 - 34}{6 - 4}=\frac{12}{2}=6\)

Step1: Use the slope - intercept form

The slope - intercept form of a line is \(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept (processing fee). We know \(m = 6\) from part 1. Let's use the point \((4,34)\) (we could use any point from the table).
Substitute \(x = 4\), \(y = 34\), and \(m = 6\) into \(y=mx + b\):
\(34=6\times4 + b\)

Step2: Solve for \(b\)

First, calculate \(6\times4 = 24\). Then the equation becomes \(34=24 + b\).
Subtract 24 from both sides: \(b=34 - 24 = 10\)

Answer:

The cost per foam hand is \(\$6\).

Part 2: Processing fee (Y - Intercept)