To solve this problem, let's break it down step-by-step:
1. Understanding the Question:
- We are given that Maria plotted the temperature [tex]\( B(t) \)[/tex] over time [tex]\( t \)[/tex].
- The average rate of change of the temperature over the interval from [tex]\( t=3 \)[/tex] to [tex]\( t=7 \)[/tex] is 8 degrees.
2. Definition of Average Rate of Change:
- The average rate of change of a function over an interval [tex]\([a, b]\)[/tex] is calculated using the formula:
[tex]\[
\frac{B(b) - B(a)}{b - a}
\][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the endpoints of the interval.
3. Applying the Average Rate of Change Formula:
- Here, [tex]\( t = 3 \)[/tex] corresponds to our [tex]\( a \)[/tex] and [tex]\( t = 7 \)[/tex] corresponds to our [tex]\( b \)[/tex].
- The rate of change over the interval from 3 to 7 is given as 8.
- This can be written as:
[tex]\[
\frac{B(7) - B(3)}{7 - 3} = 8
\][/tex]
4. Solving for the Change in Temperature:
- Simplify the denominator:
[tex]\[
7 - 3 = 4
\][/tex]
- So we have:
[tex]\[
\frac{B(7) - B(3)}{4} = 8
\][/tex]
- To find the change in temperature, multiply both sides by 4:
[tex]\[
B(7) - B(3) = 8 \times 4
\][/tex]
[tex]\[
B(7) - B(3) = 32
\][/tex]
5. Conclusion:
- The difference in the temperature between [tex]\( t = 7 \)[/tex] and [tex]\( t = 3 \)[/tex] is 32 degrees.
- Therefore, the correct statement is:
[tex]\[
\text{The temperature was 32 degrees higher when } t=7 \text{ than when } t=3.
\][/tex]
Thus, the correct statement is:
[tex]\[
\boxed{\text{The temperature was 32 degrees higher when } t=7 \text{ than when } t=3.}
\][/tex]
