one of these scenarios represents a direct variation, and the other does not. how can you tell the…

Explanation:

Step1: Recall direct - variation formula

For direct variation, the relationship is $y = kx$, where $k$ is the constant of variation.

Step2: Analyze pizza - price scenario

Let $x$ be the number of toppings and $y$ be the price. For a plain pizza ($x = 0$), $y=15$. For a pizza with 4 toppings ($x = 4$), $y = 20$. Calculate the rate of change: $\frac{y_2 - y_1}{x_2 - x_1}=\frac{20 - 15}{4-0}=\frac{5}{4}=1.25$. But when $x = 0$, $y\neq0$. So, the price of the pizza does not vary directly with the number of toppings.

Step3: Analyze marathon - running scenario

Let $x$ be the number of miles and $y$ be the time. For the first case, $x_1 = 5$ and $y_1 = 40$, so the rate is $\frac{y_1}{x_1}=\frac{40}{5}=8$. For the second case, $x_2 = 15$ and $y_2 = 120$, and $\frac{y_2}{x_2}=\frac{120}{15}=8$. Also, when $x = 0$, $y = 0$ (if you run 0 miles, it takes 0 minutes). So, the time varies directly with the number of miles.

Answer:

a. The price of the pizza does not vary directly with the number of toppings. b. The time varies directly with the number of miles.