Question

an online company sells watches with different prices. the ordering system uses the equation ( c = p + 7.50 ) to determine the total cost, ( c ), including shipping for a watch with a price of ( p ) dollars.

a. identify the independent variable and the dependent variable.

b. use the equation to complete the table.

price of watch ($), ( p )	14.00		21.00

c. how does the total cost change when the price of a watch decreases by $1? show your work.

Explanation:

Part (a)

Step1: Recall variable definitions

In a relationship \( c = p + 7.50 \), the independent variable is the one that can be changed freely, and the dependent variable depends on it. Here, \( p \) (price of watch) is independent as it can vary, and \( c \) (total cost) depends on \( p \) (since \( c \) is calculated from \( p \)).

Step1: For \( p = 14.00 \)

Use \( c = p + 7.50 \). Substitute \( p = 14.00 \):
\( c = 14.00 + 7.50 = 21.50 \)

Step2: For \( c = 26.50 \)

Solve \( 26.50 = p + 7.50 \) for \( p \). Subtract 7.50:
\( p = 26.50 - 7.50 = 19.00 \)

Step3: For \( p = 21.00 \)

Substitute \( p = 21.00 \) into \( c = p + 7.50 \):
\( c = 21.00 + 7.50 = 28.50 \)

Step4: For \( c = 35.50 \)

Solve \( 35.50 = p + 7.50 \) for \( p \). Subtract 7.50:
\( p = 35.50 - 7.50 = 28.00 \)

Step1: Let original \( p = p_1 \), new \( p = p_2 = p_1 - 11 \)

Original cost: \( c_1 = p_1 + 7.50 \)
New cost: \( c_2 = p_2 + 7.50 = (p_1 - 11) + 7.50 \)

Step2: Find \( c_1 - c_2 \)

\( c_1 - c_2 = (p_1 + 7.50) - [(p_1 - 11) + 7.50] \)
Simplify: \( p_1 + 7.50 - p_1 + 11 - 7.50 = 11 \)
So total cost decreases by $11.

Answer:

Independent variable: \( p \) (price of watch), Dependent variable: \( c \) (total cost)

Part (b)