To analyze the given problem and understand which statement must be true, let's break it down step by step.
We're given:
1. The average rate of change in the graph [tex]\( B(t) \)[/tex] over the interval from [tex]\( t = 3 \)[/tex] to [tex]\( t = 7 \)[/tex] is 8. This means the rate at which the temperature changes is 8 degrees per unit time (here, per unit of [tex]\( t \)[/tex]).
The average rate of change formula can be expressed as:
[tex]\[
\text{Rate of change} = \frac{\Delta B}{\Delta t}
\][/tex]
Where:
- [tex]\( \Delta B \)[/tex] is the change in temperature,
- [tex]\( \Delta t \)[/tex] is the change in time.
Here, [tex]\( \Delta t \)[/tex] is [tex]\( t_{\text{end}} - t_{\text{start}} \)[/tex]. Given:
[tex]\[
t_{\text{start}} = 3 \quad \text{and} \quad t_{\text{end}} = 7
\][/tex]
So,
[tex]\[
\Delta t = 7 - 3 = 4
\][/tex]
Given the rate of change is 8, we can write the equation:
[tex]\[
8 = \frac{\Delta B}{4}
\][/tex]
To find [tex]\( \Delta B \)[/tex], we rearrange the formula:
[tex]\[
\Delta B = 8 \times 4 = 32
\][/tex]
So, the change in temperature ([tex]\( \Delta B \)[/tex]) over the interval from [tex]\( t = 3 \)[/tex] to [tex]\( t = 7 \)[/tex] is 32 degrees.
This leads us to conclude that the temperature was 32 degrees higher when [tex]\( t = 7 \)[/tex] than when [tex]\( t = 3 \)[/tex].
Therefore, the correct statement must be:
- The temperature was 32 degrees higher when [tex]\( t = 7 \)[/tex] than when [tex]\( t = 3 \)[/tex].
