Question

this question has two parts. first, answer part a. then, answer part b.
part a
correct! 1 out of 1 points.
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, where x represents the number of hours and y represents the total cost.
find and interpret the rate of change.
the rate of change is
field 1
, so the hourly cost is $
field 2
.
find and interpret the initial value. express your answers as decimals.
the initial value is 3.5
field 3
, so the skate rental fee is $
field 4
.
part b
incorrect
2 tries left. please try again.
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.
y = 3x + 3.50

Explanation:

Part A: Rate of Change and Initial Value

Step 1: Calculate Rate of Change

The two points are \((x_1, y_1) = (2, 9.50)\) and \((x_2, y_2) = (5, 18.50)\). The rate of change \(m\) is given by the slope formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{18.50 - 9.50}{5 - 2} = \frac{9.00}{3} = 3
\]
So the hourly cost (rate of change) is \(\$3\) per hour.

Step 2: Find Initial Value (b)

Use the slope-intercept form \(y = mx + b\) and substitute \(m = 3\), \(x = 2\), \(y = 9.50\):
\[
9.50 = 3(2) + b \implies 9.50 = 6 + b \implies b = 9.50 - 6 = 3.50
\]
The initial value (skate rental fee) is \(\$3.50\).

Part B: Equation of the Function

The slope \(m = 3\) (hourly rate) and initial value \(b = 3.50\) (rental fee). Substitute into \(y = mx + b\):
\[
y = 3x + 3.50
\]

Final Answers

• Part A Rate of Change: \(3\) (hourly cost \(\$3\))
• Part A Initial Value: \(3.5\) (rental fee \(\$3.50\))
• Part B Equation: \(y = 3x + 3.50\)

Answer:

(Part B Equation):
\(y = 3x + 3.50\)