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 y - intercept definition

The y - intercept of a linear function \(y = mx + b\) (where \(m\) is the slope and \(b\) is the y - intercept) occurs when \(x = 0\).

Step2: Analyze the given function

In the function \(y=7000 - 175x\), \(x\) 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 increase above \(40^{\circ}\text{F}\)).

Step3: Calculate y when x = 0

Substitute \(x = 0\) into the function \(y=7000-175x\). We get \(y=7000 - 175\times0=7000\). So 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 each option:

• Option 1: When \(x = 0\), \(y = 7000

eq6825\), so this is incorrect.

• Option 2: The slope \(m=- 175\) represents the rate of change. The slope is - 175, not - 40, so this is incorrect.
• Option 3: This describes the slope (rate of change), not the y - intercept, so this is incorrect.
• Option 4: When \(x = 0\) (temperature is \(40^{\circ}\text{F}\)), \(y = 7000\), which means the coffee shop sells 7000 hot drinks when the temperature is \(40^{\circ}\text{F}\), so this is correct.

Answer:

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