Question

this question has two parts. first, answer part a. then, answer part b.
part a
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 \boxed{}, so the hourly cost is $ \boxed{}.
find and interpret the initial value. express your answers as decimals.
the initial value is \boxed{}, so the skate rental fee is $ \boxed{}.
part b
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:

Part A

Step 1: Find the rate of change (slope)

We have two points: \((2, 9.50)\) and \((5, 18.50)\). The formula for slope \(m\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
Substitute \(x_1 = 2\), \(y_1 = 9.50\), \(x_2 = 5\), \(y_2 = 18.50\) into the formula:
\(m=\frac{18.50 - 9.50}{5 - 2}=\frac{9}{3}=3\)
So the rate of change is \(3\), meaning the hourly cost is \(\$3\) per hour.

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

Use the slope - intercept form \(y = mx + b\) and one of the points, say \((2, 9.50)\) with \(m = 3\).
Substitute \(x = 2\), \(y = 9.50\) and \(m = 3\) into \(y=mx + b\):
\(9.50=3\times2 + b\)
\(9.50 = 6 + b\)
Subtract \(6\) from both sides: \(b=9.50 - 6 = 3.50\)
So the initial value is \(3.50\), meaning the skate rental fee is \(\$3.50\).

Part B

Step 1: Write the equation

We know that \(m = 3\) (rate of change) and \(b = 3.50\) (initial value) from the slope - intercept form \(y=mx + b\).
Substitute \(m = 3\) and \(b = 3.50\) into the formula:
\(y = 3x+3.5\)

Final Answers

Part A

• Rate of change: \(3\), Hourly cost: \(\$3\)
• Initial value: \(3.50\), Skate rental fee: \(\$3.50\)

Part B

The equation of the function is \(y = 3x + 3.5\)

Answer:

Part A

Step 1: Find the rate of change (slope)

We have two points: \((2, 9.50)\) and \((5, 18.50)\). The formula for slope \(m\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
Substitute \(x_1 = 2\), \(y_1 = 9.50\), \(x_2 = 5\), \(y_2 = 18.50\) into the formula:
\(m=\frac{18.50 - 9.50}{5 - 2}=\frac{9}{3}=3\)
So the rate of change is \(3\), meaning the hourly cost is \(\$3\) per hour.

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

Use the slope - intercept form \(y = mx + b\) and one of the points, say \((2, 9.50)\) with \(m = 3\).
Substitute \(x = 2\), \(y = 9.50\) and \(m = 3\) into \(y=mx + b\):
\(9.50=3\times2 + b\)
\(9.50 = 6 + b\)
Subtract \(6\) from both sides: \(b=9.50 - 6 = 3.50\)
So the initial value is \(3.50\), meaning the skate rental fee is \(\$3.50\).

Part B

Step 1: Write the equation

We know that \(m = 3\) (rate of change) and \(b = 3.50\) (initial value) from the slope - intercept form \(y=mx + b\).
Substitute \(m = 3\) and \(b = 3.50\) into the formula:
\(y = 3x+3.5\)

Final Answers

Part A

• Rate of change: \(3\), Hourly cost: \(\$3\)
• Initial value: \(3.50\), Skate rental fee: \(\$3.50\)

Part B

The equation of the function is \(y = 3x + 3.5\)