Question

the amount of hot drinks, y, a coffee shop sells as the temperature increases x degrees above 40°f can be modeled by the function y = 7000 - 175x. which sentence correctly interprets the y-intercept in the context of this problem? the coffee shop sells 6825 hot drinks when the temperature is 40°f. the number of hot drinks sold decreases by 40 for each degree the temperature increases above 40°f. the number of hot drinks sold decreases by 175 for each degree the temperature increases above 40°f. the coffee shop sells 7000 hot drinks when the temperature is 40°f.

Explanation:

Step1: Recall the slope - intercept form of a linear equation

The slope - intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the $y$ - intercept. In the given equation $y=7000 - 175x$, we can rewrite it as $y=- 175x + 7000$. So, $b = 7000$ (the $y$ - intercept) and $m=-175$ (the slope).

Step2: Interpret the $y$ - intercept in the context of the problem

The variable $x$ represents the number of degrees above $40^{\circ}\text{F}$. When $x = 0$, it means the temperature is $40^{\circ}\text{F}$ (since $x$ is the number of degrees above $40^{\circ}\text{F}$). To find the value of $y$ (the number of hot drinks sold) when $x = 0$, we substitute $x = 0$ into the equation $y=7000-175x$.
Substituting $x = 0$ into the equation: $y=7000-175\times0=7000$. This means that when the temperature is $40^{\circ}\text{F}$ (i.e., $x = 0$), the number of hot drinks sold $y$ is 7000.

Now let's analyze the other options:

• Option 1: When $x = 0$, $y = 7000

eq6825$, so this option is incorrect.

• Option 2: The slope $m=- 175$ means that for each degree increase in temperature above $40^{\circ}\text{F}$ (i.e., for each increase in $x$ by 1), $y$ decreases by 175, not 40. So this option is incorrect.
• Option 3: The slope $m = - 175$ represents the rate of change of $y$ with respect to $x$, not the $y$ - intercept. So this option is incorrect.

Answer:

The coffee shop sells 7000 hot drinks when the temperature is $40^{\circ}\text{F}$. (The last option: "The coffee shop sells 7000 hot drinks when the temperature is $40^{\circ}\text{F}$")