Question

1. a function ( k(x) ) has an average rate of change equal to 7 on the interval ( 5leq xleq9 ). if ( k(5)=12 ), then which of the following must be the value of ( k(9) )?

(1) 28
(2) 36
(3) 40
(4) 52

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \(y = k(x)\) over the interval \([a,b]\) is given by \(\frac{k(b)-k(a)}{b - a}\). Here, \(a = 5\), \(b=9\), and the average rate of change is \(7\), and \(k(5)=12\).

Step2: Substitute the values into the formula

We have \(\frac{k(9)-k(5)}{9 - 5}=7\). Substitute \(k(5) = 12\) into the equation: \(\frac{k(9)-12}{4}=7\).

Step3: Solve for \(k(9)\)

Multiply both sides of the equation \(\frac{k(9)-12}{4}=7\) by \(4\): \(k(9)-12=7\times4\). Then \(k(9)-12 = 28\). Add \(12\) to both sides: \(k(9)=28 + 12\).

Answer:

\(k(9)=40\), so the answer is (3).