Question
A dance studio in Charleston, South Carolina, charges a one-time registration fee of $46.25 and $35 per month for classes. (6 points) Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse. Part A: What is the equation for the cost of dance classes if y is the total cost (in dollars) for joining for a certain number of months,x y= Response area Part B: Graph the equation using Desmos. (Link to Desmos) Part C: Determine the key feature of the function x-intercept Response area y-intercept Response area slope Response area domain In the context of the problem Response area range in the context of the problem Response area
Answers
GPT-4o mini
Let's break it down step-by-step for each part of your question.
**Part A: Equation for the total cost of dance classes**
1. Registration fee: $46.25 (one-time).
2. Monthly fee: $35 per month (for a certain number of months, which we will denote as \( x \)).
To write an equation for the total cost \( y \), we have:
\[
y = 35x + 46.25
\]
Where:
- \( y \) is the total cost,
- \( x \) is the number of months.
**Part B: Graphing the equation using Desmos**
To graph the equation in Desmos:
1. Go to [Desmos](https://www.desmos.com/calculator).
2. Input the equation: \( y = 35x + 46.25 \).
3. You will see a straight line representing the cost of classes as the number of months increases.
**Part C: Key features of the function**
1. **X-intercept**: This is where the graph crosses the x-axis (\( y = 0 \)).
- To find this, set \( y = 0 \):
\[
0 = 35x + 46.25 \\
35x = -46.25 \\
x = -\frac{46.25}{35} \approx -1.321
\]
So, the x-intercept (in the context of the problem, it's not meaningful since you cannot join for a negative number of months).
2. **Y-intercept**: This is where the graph crosses the y-axis (\( x = 0 \)).
- For \( x = 0 \):
\[
y = 35(0) + 46.25 = 46.25
\]
So, the y-intercept is \( 46.25 \).
3. **Slope**: The slope is the coefficient of \( x \).
\[
\text{slope} = 35
\]
This means for every additional month, the cost increases by $35.
4. **Domain**: The domain is the set of values that \( x \) can take in the context of the problem.
- In the context of the problem, \( x \geq 0 \) (you cannot take classes for a negative number of months, so the domain is \( [0, \infty) \)).
5. **Range**: The range is the set of possible values for \( y \).
- In the context of the problem, since \( y \) starts at 46.25 and increases with every month, the range is \( [46.25, \infty) \).
### Summary of Responses
- **Part A:** \( y = 35x + 46.25 \)
- **X-intercept:** Approximately \(-1.321\) (not meaningful in context)
- **Y-intercept:** \( 46.25 \)
- **Slope:** \( 35 \)
- **Domain (in context):** \( [0, \infty) \)
- **Range (in context):** \( [46.25, \infty) \)
You can fill these responses in the respective areas from your question format.
**Part A: Equation for the total cost of dance classes**
1. Registration fee: $46.25 (one-time).
2. Monthly fee: $35 per month (for a certain number of months, which we will denote as \( x \)).
To write an equation for the total cost \( y \), we have:
\[
y = 35x + 46.25
\]
Where:
- \( y \) is the total cost,
- \( x \) is the number of months.
**Part B: Graphing the equation using Desmos**
To graph the equation in Desmos:
1. Go to [Desmos](https://www.desmos.com/calculator).
2. Input the equation: \( y = 35x + 46.25 \).
3. You will see a straight line representing the cost of classes as the number of months increases.
**Part C: Key features of the function**
1. **X-intercept**: This is where the graph crosses the x-axis (\( y = 0 \)).
- To find this, set \( y = 0 \):
\[
0 = 35x + 46.25 \\
35x = -46.25 \\
x = -\frac{46.25}{35} \approx -1.321
\]
So, the x-intercept (in the context of the problem, it's not meaningful since you cannot join for a negative number of months).
2. **Y-intercept**: This is where the graph crosses the y-axis (\( x = 0 \)).
- For \( x = 0 \):
\[
y = 35(0) + 46.25 = 46.25
\]
So, the y-intercept is \( 46.25 \).
3. **Slope**: The slope is the coefficient of \( x \).
\[
\text{slope} = 35
\]
This means for every additional month, the cost increases by $35.
4. **Domain**: The domain is the set of values that \( x \) can take in the context of the problem.
- In the context of the problem, \( x \geq 0 \) (you cannot take classes for a negative number of months, so the domain is \( [0, \infty) \)).
5. **Range**: The range is the set of possible values for \( y \).
- In the context of the problem, since \( y \) starts at 46.25 and increases with every month, the range is \( [46.25, \infty) \).
### Summary of Responses
- **Part A:** \( y = 35x + 46.25 \)
- **X-intercept:** Approximately \(-1.321\) (not meaningful in context)
- **Y-intercept:** \( 46.25 \)
- **Slope:** \( 35 \)
- **Domain (in context):** \( [0, \infty) \)
- **Range (in context):** \( [46.25, \infty) \)
You can fill these responses in the respective areas from your question format.