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?
A.
The initial height of the sculpture is 2 inches.
B.
The sculpture melts 2 inches each hour.
C.
It takes 2 hours for the sculpture to melt completely.
D.
It takes the sculpture 24 hours to melt completely.
Answers
GPT-4o mini
To interpret the function \( h(t) = -2t + 24 \), we can analyze the components of the equation:
1. The equation is in the form \( h(t) = mt + b \), where \( m \) is the slope and \( b \) is the y-intercept.
2. In this case, \( m = -2 \) and \( b = 24 \).
- The **y-intercept (24)** represents the **initial height** of the sculpture when \( t = 0 \). This means that the initial height is 24 inches.
- The **slope (-2)** indicates that for every 1 hour that passes (\( t \) increases by 1), the height \( h(t) \) decreases by 2 inches. Therefore, the sculpture melts 2 inches each hour.
Now, we can evaluate the answer choices:
A. **The initial height of the sculpture is 2 inches.**
- This is incorrect. The initial height is actually 24 inches.
B. **The sculpture melts 2 inches each hour.**
- This is correct. The slope of -2 indicates that the height decreases by 2 inches for each hour.
C. **It takes 2 hours for the sculpture to melt completely.**
- This is incorrect. To find out how long it takes for the sculpture to melt completely, we solve for \( t \) when \( h(t) = 0 \):
\[
0 = -2t + 24 \implies 2t = 24 \implies t = 12 \text{ hours}.
\]
So, it takes 12 hours to melt completely, not 2 hours.
D. **It takes the sculpture 24 hours to melt completely.**
- This is also incorrect. As calculated above, it takes 12 hours to melt completely, not 24 hours.
Thus, the correct interpretation of the function is:
**B. The sculpture melts 2 inches each hour.**
1. The equation is in the form \( h(t) = mt + b \), where \( m \) is the slope and \( b \) is the y-intercept.
2. In this case, \( m = -2 \) and \( b = 24 \).
- The **y-intercept (24)** represents the **initial height** of the sculpture when \( t = 0 \). This means that the initial height is 24 inches.
- The **slope (-2)** indicates that for every 1 hour that passes (\( t \) increases by 1), the height \( h(t) \) decreases by 2 inches. Therefore, the sculpture melts 2 inches each hour.
Now, we can evaluate the answer choices:
A. **The initial height of the sculpture is 2 inches.**
- This is incorrect. The initial height is actually 24 inches.
B. **The sculpture melts 2 inches each hour.**
- This is correct. The slope of -2 indicates that the height decreases by 2 inches for each hour.
C. **It takes 2 hours for the sculpture to melt completely.**
- This is incorrect. To find out how long it takes for the sculpture to melt completely, we solve for \( t \) when \( h(t) = 0 \):
\[
0 = -2t + 24 \implies 2t = 24 \implies t = 12 \text{ hours}.
\]
So, it takes 12 hours to melt completely, not 2 hours.
D. **It takes the sculpture 24 hours to melt completely.**
- This is also incorrect. As calculated above, it takes 12 hours to melt completely, not 24 hours.
Thus, the correct interpretation of the function is:
**B. The sculpture melts 2 inches each hour.**