Question

this question has two parts. first, answer part a. then, answer part b.
part a
the table shows the distance penelope is from the park as she walks to soccer practice. assume the relationship between the two quantities is linear.

time (min), x	5	10	15	20

find and interpret the rate of change.
the rate of change is \boxed{}, so penelope is \boxed{} meters closer to the park every minute.
find and interpret the initial value.
the initial value is \boxed{}, so penelope was initially \boxed{} meters from the park.
part b
write the equation in the form $y = mx + b$.

Explanation:

Part A: Rate of Change and Initial Value

Step 1: Calculate Rate of Change

The rate of change (slope \( m \)) for a linear relationship is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Let's use two points, e.g., \( (x_1, y_1) = (5, 1930) \) and \( (x_2, y_2) = (10, 1380) \).

$$ m = \frac{1380 - 1930}{10 - 5} = \frac{-550}{5} = -110 $$

(We can verify with other points: \( (10, 1380) \) and \( (15, 830) \): \( \frac{830 - 1380}{15 - 10} = \frac{-550}{5} = -110 \), consistent.)

Step 2: Interpret Rate of Change

The rate of change is \( -110 \), meaning Penelope’s distance from the park decreases by 110 meters per minute (she gets 110 meters closer each minute).

Step 3: Find Initial Value (\( b \) in \( y = mx + b \))

Use point \( (5, 1930) \) and \( m = -110 \):

$$ 1930 = -110(5) + b \implies 1930 = -550 + b \implies b = 1930 + 550 = 2480 $$

Interpretation: The initial value (when \( x = 0 \), time = 0) is 2480 meters, so Penelope was initially 2480 meters from the park.

Part B: Equation in \( y = mx + b \)

From Part A, \( m = -110 \) and \( b = 2480 \). Thus:

$$ y = -110x + 2480 $$

Final Answers (Part A)

• Rate of change: \( \boldsymbol{-110} \) (so Penelope is 110 meters closer per minute).
• Initial value: \( \boldsymbol{2480} \) (so Penelope was initially 2480 meters from the park).

Final Answer (Part B)

$$ \boldsymbol{y = -110x + 2480} $$

Answer:

Part A: Rate of Change and Initial Value

Step 1: Calculate Rate of Change

The rate of change (slope \( m \)) for a linear relationship is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Let's use two points, e.g., \( (x_1, y_1) = (5, 1930) \) and \( (x_2, y_2) = (10, 1380) \).

$$ m = \frac{1380 - 1930}{10 - 5} = \frac{-550}{5} = -110 $$

(We can verify with other points: \( (10, 1380) \) and \( (15, 830) \): \( \frac{830 - 1380}{15 - 10} = \frac{-550}{5} = -110 \), consistent.)

Step 2: Interpret Rate of Change

The rate of change is \( -110 \), meaning Penelope’s distance from the park decreases by 110 meters per minute (she gets 110 meters closer each minute).

Step 3: Find Initial Value (\( b \) in \( y = mx + b \))

Use point \( (5, 1930) \) and \( m = -110 \):

$$ 1930 = -110(5) + b \implies 1930 = -550 + b \implies b = 1930 + 550 = 2480 $$

Interpretation: The initial value (when \( x = 0 \), time = 0) is 2480 meters, so Penelope was initially 2480 meters from the park.

Part B: Equation in \( y = mx + b \)

From Part A, \( m = -110 \) and \( b = 2480 \). Thus:

$$ y = -110x + 2480 $$

Final Answers (Part A)

• Rate of change: \( \boldsymbol{-110} \) (so Penelope is 110 meters closer per minute).
• Initial value: \( \boldsymbol{2480} \) (so Penelope was initially 2480 meters from the park).

Final Answer (Part B)

$$ \boldsymbol{y = -110x + 2480} $$