Question

maria created a graph of ( b(t) ), the temperature over time. for the interval between ( t = 3 ) and ( t = 7 ), the average rate of change in her graph of ( b(t) ) is 8. which statement must be true?
the temperature was 8 degrees higher when ( t = 7 ) than when ( t = 3 ).
the temperature was 8 times higher when ( t = 7 ) than when ( t = 3 ).
the temperature was 32 degrees higher when ( t = 7 ) than when ( t = 3 ).
the temperature was 2 degrees higher when ( t = 7 ) than when ( t = 3 ).

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = B(t)\) over the interval \([a,b]\) is \(\frac{B(b)-B(a)}{b - a}\). Here, \(a = 3\), \(b=7\), and the average rate of change is \(8\).

Step2: Substitute values into the formula

Substitute into \(\frac{B(7)-B(3)}{7 - 3}=8\).

Step3: Simplify the denominator

Since \(7-3 = 4\), the equation becomes \(\frac{B(7)-B(3)}{4}=8\).

Step4: Solve for \(B(7)-B(3)\)

Multiply both sides of the equation \(\frac{B(7)-B(3)}{4}=8\) by \(4\). We get \(B(7)-B(3)=8\times4\).

Step5: Calculate the product

\(8\times4=32\), so \(B(7)-B(3) = 32\). This means the temperature was \(32\) degrees higher when \(t = 7\) than when \(t=3\).

Answer:

The temperature was 32 degrees higher when \(t = 7\) than when \(t = 3\).