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)

Explanation:

Part 1: How much do they charge per foam hand? (Slope)

Step1: Recall slope formula

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let's take two points from the table, e.g., \((4, 34)\) and \((6, 46)\).

Step2: Calculate the slope

Substitute \( x_1 = 4, y_1 = 34, x_2 = 6, y_2 = 46 \) into the formula:
\( m=\frac{46 - 34}{6 - 4}=\frac{12}{2}=6 \).
We can verify with another pair, like \((8, 58)\) and \((10, 70)\): \( \frac{70 - 58}{10 - 8}=\frac{12}{2}=6 \). So the slope (cost per foam hand) is 6.

Step1: Use slope-intercept form \( y = mx + b \)

We know \( m = 6 \) (from part 1), and we can use a point, e.g., \((4, 34)\), where \( x = 4 \), \( y = 34 \).

Step2: Solve for \( b \) (y-intercept)

Substitute into \( y = mx + b \): \( 34 = 6(4) + b \).
Calculate \( 6(4)=24 \), so \( 34 = 24 + b \).
Subtract 24 from both sides: \( b = 34 - 24 = 10 \).
We can check with another point, e.g., \((6, 46)\): \( 46 = 6(6) + b \Rightarrow 46 = 36 + b \Rightarrow b = 10 \). So the y-intercept (processing fee) is 10.

Answer:

6

Part 2: How much is the processing fee? (Y-intercept)