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Answer :
Sure! Let's work through the problem step-by-step:
### Step 1: Determine Which Equation Models the Data
We are given different models to choose from that describe how the temperature is changing over time. The data in the table shows a gradual decrease in temperature, which suggests an exponential decay model. Let's evaluate the models provided:
1. [tex]\(40(0.98)^t\)[/tex]: This equation represents exponential decay, and we will check if it fits the data.
2. [tex]\(40^2\)[/tex]: This is not an exponential model; it's a constant and does not depend on [tex]\(t\)[/tex] (time).
3. [tex]\(40(0.8)^t\)[/tex]: Another exponential decay model but with a faster rate of decay than the first.
4. [tex]\(40(1.02)^t\)[/tex]: This represents exponential growth, which does not fit the cooling process.
Let's try the model [tex]\(40(0.98)^t\)[/tex]:
- At [tex]\(t = 1\)[/tex]: [tex]\(40(0.98)^1 = 39.2\)[/tex]
The model [tex]\(40(0.98)^t\)[/tex] closely matches the observed data at [tex]\(t = 1\)[/tex], which is in line with the recorded temperature of 39.2°F. Therefore, the appropriate model is [tex]\(40(0.98)^t\)[/tex].
### Step 2: Calculate the Temperature After 11 Hours
Now that we have determined that the temperature follows the model [tex]\(40(0.98)^t\)[/tex], we can find the temperature after 11 hours using this equation.
- Plug [tex]\(t = 11\)[/tex] into the model:
[tex]\[
T = 40 \times (0.98)^{11}
\][/tex]
When you calculate this, you find that the temperature after 11 hours is approximately 32.0°F when rounded to the nearest tenth.
This gives us the solution to the problem based on the model and data provided.
### Step 1: Determine Which Equation Models the Data
We are given different models to choose from that describe how the temperature is changing over time. The data in the table shows a gradual decrease in temperature, which suggests an exponential decay model. Let's evaluate the models provided:
1. [tex]\(40(0.98)^t\)[/tex]: This equation represents exponential decay, and we will check if it fits the data.
2. [tex]\(40^2\)[/tex]: This is not an exponential model; it's a constant and does not depend on [tex]\(t\)[/tex] (time).
3. [tex]\(40(0.8)^t\)[/tex]: Another exponential decay model but with a faster rate of decay than the first.
4. [tex]\(40(1.02)^t\)[/tex]: This represents exponential growth, which does not fit the cooling process.
Let's try the model [tex]\(40(0.98)^t\)[/tex]:
- At [tex]\(t = 1\)[/tex]: [tex]\(40(0.98)^1 = 39.2\)[/tex]
The model [tex]\(40(0.98)^t\)[/tex] closely matches the observed data at [tex]\(t = 1\)[/tex], which is in line with the recorded temperature of 39.2°F. Therefore, the appropriate model is [tex]\(40(0.98)^t\)[/tex].
### Step 2: Calculate the Temperature After 11 Hours
Now that we have determined that the temperature follows the model [tex]\(40(0.98)^t\)[/tex], we can find the temperature after 11 hours using this equation.
- Plug [tex]\(t = 11\)[/tex] into the model:
[tex]\[
T = 40 \times (0.98)^{11}
\][/tex]
When you calculate this, you find that the temperature after 11 hours is approximately 32.0°F when rounded to the nearest tenth.
This gives us the solution to the problem based on the model and data provided.
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