Question

lesson 16 | session 2
3 each day darnell buys a cup of soup and a salad for lunch. a cup of soup costs $3.75 and the salad costs a certain amount per ounce. the equation below models the total cost of darnell’s lunch.
y = 0.45x + 3.75
a. what do the variables x and y represent? use the phrase is a function of to describe how the equation relates these quantities to one another.
b. what does the value of the function for x = 0 represent?
c. what does the rate of change represent?
d. what is the cost of an 8-ounce salad without soup? how do you know?
4 neta’s family pays an annual membership fee to visit an aquarium. they pay $15 for parking each time they visit the aquarium. the equation y = 15x + 125 represents their total yearly cost in dollars. which statement about the function is true? select all the correct answers.
a the initial value is 15.
b x represents the cost of parking each time they visit.
c the rate of change is 15.
d the initial value represents the annual membership fee.
e the number of times they visit is a function of the total yearly cost.
f the total yearly cost is a function of the number of times they park.

Explanation:

Problem 3

Part a

Step1: Analyze the equation structure

The equation is in the form \( y = mx + b \), where \( b \) is the fixed cost and \( mx \) is the variable cost. The soup costs a fixed $3.75, and the salad costs $0.45 per ounce.

Step2: Define variables

So, \( x \) represents the number of ounces of the salad (since the cost per ounce is 0.45, multiplying by \( x \) gives the salad's cost), and \( y \) represents the total cost of Darnell's lunch (soup + salad). The total cost \( y \) is a function of the number of ounces of salad \( x \), meaning \( y \) depends on \( x \) as \( y = 0.45x + 3.75 \).

Step1: Substitute \( x = 0 \) into the equation

Substitute \( x = 0 \) into \( y = 0.45x + 3.75 \).

Step2: Calculate and interpret

When \( x = 0 \), \( y = 0.45(0)+ 3.75 = 3.75 \). Since \( x \) is the number of ounces of salad, \( x = 0 \) means no salad is bought. So the value of the function at \( x = 0 \) represents the cost of the lunch when only the soup is bought (no salad), which is the cost of the soup.

Step1: Recall the slope - rate of change relationship

In the linear equation \( y = mx + b \), \( m \) is the rate of change (slope). Here, the equation is \( y = 0.45x + 3.75 \), so the rate of change is 0.45.

Step2: Interpret the rate of change

Since \( x \) is the number of ounces of salad and the coefficient of \( x \) is 0.45, the rate of change (0.45) represents the cost per ounce of the salad (in dollars per ounce).

Answer:

\( x \) represents the number of ounces of the salad; \( y \) represents the total cost of Darnell’s lunch. The total cost of Darnell’s lunch (\( y \)) is a function of the number of ounces of the salad (\( x \)), related by \( y = 0.45x + 3.75 \).

Part b