Question
An ice sculpture is used as a centerpiece at a banquet. Once the sculpture is removed from the freezer it begins to melt. The height of the sculpture can be represented by the function h(t)=−2t+24
, as shown on the graph.
Which of the following statements is a correct interpretation of the function representing the height of the sculpture? Select all that apply.
(1 point)
Responses
The initial height of the sculpture is 24 inches.
The initial height of the sculpture is 24 inches.
The initial height of the sculpture is 2 inches.
The initial height of the sculpture is 2 inches.
The sculpture melts 2 inches each hour.
The sculpture melts 2 inches each hour.
It takes the sculpture 24 hours to melt completely.
It takes the sculpture 24 hours to melt completely.
It takes 2 hours for the sculpture to melt completely.
It takes 2 hours for the sculpture to melt completely.
The sculpture melts 24 inches each hour.
Answers
GPT-4o mini
To interpret the function \( h(t) = -2t + 24 \), we can analyze it as follows:
1. **Initial height**: When \( t = 0 \), the height \( h(0) = -2(0) + 24 = 24 \) inches. This statement is correct.
- **Response**: The initial height of the sculpture is 24 inches. **(Correct)**
2. **Height decrease**: The term \(-2t\) indicates that the height decreases by 2 inches for each hour that passes. This statement is correct.
- **Response**: The sculpture melts 2 inches each hour. **(Correct)**
3. **Time to melt completely**: To find out when the sculpture melts completely, set \( h(t) = 0 \):
\[
0 = -2t + 24
\]
\[
2t = 24 \quad \Rightarrow \quad t = 12
\]
This indicates that it takes 12 hours for the sculpture to melt completely, not 2 or 24 hours. Therefore, both statements about taking 2 hours and 24 hours to melt completely are incorrect.
- **Response**: It takes the sculpture 24 hours to melt completely. **(Incorrect)**
- **Response**: It takes 2 hours for the sculpture to melt completely. **(Incorrect)**
4. **Melting rate**: The sculpture does not melt 24 inches each hour; it melts 2 inches each hour. This statement is incorrect.
- **Response**: The sculpture melts 24 inches each hour. **(Incorrect)**
In summary, the correct interpretations of the function \( h(t) = -2t + 24 \) are:
- The initial height of the sculpture is 24 inches.
- The sculpture melts 2 inches each hour.
1. **Initial height**: When \( t = 0 \), the height \( h(0) = -2(0) + 24 = 24 \) inches. This statement is correct.
- **Response**: The initial height of the sculpture is 24 inches. **(Correct)**
2. **Height decrease**: The term \(-2t\) indicates that the height decreases by 2 inches for each hour that passes. This statement is correct.
- **Response**: The sculpture melts 2 inches each hour. **(Correct)**
3. **Time to melt completely**: To find out when the sculpture melts completely, set \( h(t) = 0 \):
\[
0 = -2t + 24
\]
\[
2t = 24 \quad \Rightarrow \quad t = 12
\]
This indicates that it takes 12 hours for the sculpture to melt completely, not 2 or 24 hours. Therefore, both statements about taking 2 hours and 24 hours to melt completely are incorrect.
- **Response**: It takes the sculpture 24 hours to melt completely. **(Incorrect)**
- **Response**: It takes 2 hours for the sculpture to melt completely. **(Incorrect)**
4. **Melting rate**: The sculpture does not melt 24 inches each hour; it melts 2 inches each hour. This statement is incorrect.
- **Response**: The sculpture melts 24 inches each hour. **(Incorrect)**
In summary, the correct interpretations of the function \( h(t) = -2t + 24 \) are:
- The initial height of the sculpture is 24 inches.
- The sculpture melts 2 inches each hour.