Question

3a) a roller skating rink charges a skate rental fee and an hourly rate to skate. the total cost to skate for 2 hours is $9.50 and for 5 hours is $18.50. assume the relationship is linear.
find and interpret the rate of change.
the rate of change is _____________, so the hourly cost is $_____________.
find and interpret the initial value. express your answers as decimals.
the initial value is _____________, so the skate rental fee is $_____________.
3b) write the equation of the function in the form $y = mx + b$, where $x$ represents the number of hours and $y$ represents the total cost.

Explanation:

3A) Rate of Change and Initial Value

Step 1: Find the rate of change (slope)

The rate of change (slope \( m \)) of a linear relationship between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Here, let \( x \) be the number of hours and \( y \) be the total cost. The two points are \((2, 9.50)\) and \((5, 18.50)\).
\[
m=\frac{18.50 - 9.50}{5 - 2}=\frac{9}{3}=3
\]
The rate of change is \( 3 \) dollars per hour, so the hourly cost is \( \$3.00 \).

Step 2: Find the initial value (y-intercept \( b \))

Use the slope-intercept form \( y = mx + b \) and one of the points (e.g., \((2, 9.50)\)) to solve for \( b \). Substitute \( m = 3 \), \( x = 2 \), and \( y = 9.50 \):
\[
9.50=3\times2 + b
\]
\[
9.50 = 6 + b
\]
Subtract \( 6 \) from both sides:
\[
b=9.50 - 6 = 3.50
\]
The initial value is \( 3.50 \), so the skate rental fee is \( \$3.50 \).

3B) Equation of the Function

Step 1: Identify \( m \) and \( b \)

From part 3A, we found \( m = 3 \) (hourly rate) and \( b = 3.50 \) (rental fee).

Step 2: Write the equation

Using the slope-intercept form \( y=mx + b \), substitute \( m = 3 \) and \( b = 3.50 \):
\[
y = 3x + 3.5
\]

Final Answers

3A)

• The rate of change is \( \boldsymbol{3} \), so the hourly cost is \( \boldsymbol{\$3.00} \).
• The initial value is \( \boldsymbol{3.50} \), so the skate rental fee is \( \boldsymbol{\$3.50} \).

3B)
The equation of the function is \( \boldsymbol{y = 3x + 3.5} \).

Answer:

3A) Rate of Change and Initial Value

Step 1: Find the rate of change (slope)

The rate of change (slope \( m \)) of a linear relationship between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Here, let \( x \) be the number of hours and \( y \) be the total cost. The two points are \((2, 9.50)\) and \((5, 18.50)\).
\[
m=\frac{18.50 - 9.50}{5 - 2}=\frac{9}{3}=3
\]
The rate of change is \( 3 \) dollars per hour, so the hourly cost is \( \$3.00 \).

Step 2: Find the initial value (y-intercept \( b \))

Use the slope-intercept form \( y = mx + b \) and one of the points (e.g., \((2, 9.50)\)) to solve for \( b \). Substitute \( m = 3 \), \( x = 2 \), and \( y = 9.50 \):
\[
9.50=3\times2 + b
\]
\[
9.50 = 6 + b
\]
Subtract \( 6 \) from both sides:
\[
b=9.50 - 6 = 3.50
\]
The initial value is \( 3.50 \), so the skate rental fee is \( \$3.50 \).

3B) Equation of the Function

Step 1: Identify \( m \) and \( b \)

From part 3A, we found \( m = 3 \) (hourly rate) and \( b = 3.50 \) (rental fee).

Step 2: Write the equation

Using the slope-intercept form \( y=mx + b \), substitute \( m = 3 \) and \( b = 3.50 \):
\[
y = 3x + 3.5
\]

Final Answers

3A)

• The rate of change is \( \boldsymbol{3} \), so the hourly cost is \( \boldsymbol{\$3.00} \).
• The initial value is \( \boldsymbol{3.50} \), so the skate rental fee is \( \boldsymbol{\$3.50} \).

3B)
The equation of the function is \( \boldsymbol{y = 3x + 3.5} \).