A dog boarding facility charges daily based on a dog’s weight. If the dog weighs under 15 lbs, the rate is $30. If the dog is between 15 and 50 lbs, it is $35. If the dog is 50lbs or more, the charge is $35 plus $3 for each additional 5 lbs over 50 lbs. Which of the following best defines the quantity and scale for the x-axis for a piecewise linear graph of the fee structure?

Question

A dog boarding facility charges daily based on a dog’s weight. If the dog weighs under 15 lbs, the rate is $30. If the dog is between 15 and 50 lbs, it is $35. If the dog is 50lbs or more, the charge is $35 plus $3 for each additional 5 lbs over 50 lbs. Which of the following best defines the quantity and scale for the x-axis for a piecewise linear graph of the fee structure?
Responses:

amount; 1
amount;10

weight; 1
weight; 10

GPT-4o mini · Answer

To define the quantity and scale for the x-axis of a piecewise linear graph of the fee structure based on the dog's weight, we need to focus on the relevant variable, which is the weight of the dog.

The weight categories outlined in the boarding facility's fees are:

Under 15 lbs = $30
Between 15 lbs and 50 lbs = $35
50 lbs or more, with a base fee of $35 plus an additional $3 for every 5 lbs over 50 lbs

This indicates that the x-axis represents the weight of the dog.

Since the scale needs to be practical for the weights that are relevant (e.g., starting below 15 lbs and extending to larger weights), the best choice here would be:

weight; 10

This indicates that we are measuring the weight of the dog, and using a scale of 10 makes sense for a range that includes dogs under 15 lbs and up.

Yes. · Answer

A gas station is offering a deal. If you buy 10 gallons or more of gas, you will be charged $2.25 per gallon instead of the regular $2.50 per gallon. The deal is modeled by the following piecewise function.

f(x)={2.5x if x<102.25x if x≥10

If you were to create a graph of this piecewise function, which point would have an open circle?

Responses:

(10,22.5)

(0,0)

(1,2.5)

(10,25)

GPT-4o mini · Answer

In the given piecewise function: $$ f(x) = \begin{cases} 2.5x & \text{if } x < 10 \ 2.25x & \text{if } x \geq 10 \end{cases} $$ The point where the condition changes is at $x = 10$. 1. For $x < 10$, the value of the function is calculated using $f(x) = 2.5x$. 2. At exactly $x = 10$, we transition into the second case where $f(x) = 2.25x$. To find the value at $x = 10$ under the first condition: - When $x = 10$, $$ f(x) = 2.5 \times 10 = 25 $$ But for $x \geq 10$, the value is calculated using the other part of the piecewise function: $$ f(10) = 2.25 \times 10 = 22.5 $$ Hence, at the point $ (10, 25) $, the function takes the value of 25 in the first piece (the function does not include this point because it’s defined only for $x < 10$), and the value actually at $x=10$ using the second piece is 22.5. So, the point $ (10, 25) $ will have an open circle, representing that the function does not include this point. The correct answer is: **(10, 25)**