Maria created a graph of [tex]\( B(t) \)[/tex], the temperature over time. For the interval between [tex]\( t=3 \)[/tex] and [tex]\( t=7 \)[/tex], the average rate of change in her graph of [tex]\( B(t) \)[/tex] is 8. Which statement must be true?
A. The temperature was 8 degrees higher when [tex]\( t=7 \)[/tex] than when [tex]\( t=3 \)[/tex].
B. The temperature was 8 times higher when [tex]\( t=7 \)[/tex] than when [tex]\( t=3 \)[/tex].
C. The temperature was 32 degrees higher when [tex]\( t=7 \)[/tex] than when [tex]\( t=3 \)[/tex].
D. The temperature was 2 degrees higher when [tex]\( t=7 \)[/tex] than when [tex]\( t=3 \)[/tex].
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]
