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 its components:
1. The function's output, \( h(t) \), represents the height of the sculpture at time \( t \) (where \( t \) is in hours).
2. The term \( -2t \) indicates that the height decreases by 2 inches for each hour that passes.
3. The constant term \( 24 \) represents the initial height of the sculpture when \( t = 0 \).
Now, let's evaluate each option:
A. The initial height of the sculpture is 2 inches.
**This is incorrect.** The initial height at \( t = 0 \) is \( h(0) = -2(0) + 24 = 24 \) inches.
B. The sculpture melts 2 inches each hour.
**This is correct.** The slope of the function indicates a decrease of 2 inches per hour.
C. It takes 2 hours for the sculpture to melt completely.
**This is incorrect.** To find when the sculpture melts completely (when \( h(t) = 0 \)):
\[
0 = -2t + 24 \\
2t = 24 \\
t = 12 \text{ hours}
\]
So it takes 12 hours to melt completely, not 2.
D. It takes the sculpture 24 hours to melt completely.
**This is incorrect.** As calculated above, it takes 12 hours to melt completely.
The correct interpretation is:
**B. The sculpture melts 2 inches each hour.**
1. The function's output, \( h(t) \), represents the height of the sculpture at time \( t \) (where \( t \) is in hours).
2. The term \( -2t \) indicates that the height decreases by 2 inches for each hour that passes.
3. The constant term \( 24 \) represents the initial height of the sculpture when \( t = 0 \).
Now, let's evaluate each option:
A. The initial height of the sculpture is 2 inches.
**This is incorrect.** The initial height at \( t = 0 \) is \( h(0) = -2(0) + 24 = 24 \) inches.
B. The sculpture melts 2 inches each hour.
**This is correct.** The slope of the function indicates a decrease of 2 inches per hour.
C. It takes 2 hours for the sculpture to melt completely.
**This is incorrect.** To find when the sculpture melts completely (when \( h(t) = 0 \)):
\[
0 = -2t + 24 \\
2t = 24 \\
t = 12 \text{ hours}
\]
So it takes 12 hours to melt completely, not 2.
D. It takes the sculpture 24 hours to melt completely.
**This is incorrect.** As calculated above, it takes 12 hours to melt completely.
The correct interpretation is:
**B. The sculpture melts 2 inches each hour.**