QuestionCalculate the average rate of change of the function $k(x)=-16 \sqrt{x}$ from $x=12$ to $x=15$ .

Studdy Solution

STEP 1

Assumptions1. The function is $k(x)=-16 \sqrt{x}$ . We are looking for the average rate of change over the interval $[12,15]$

STEP 2

The average rate of change of a function over an interval $[a,b]$ is given by the formula $\text{Average rate of change} = \frac{k(b) - k(a)}{b - a}$

STEP 3

We plug in the values for $a$ and $b$ from our interval into the formula. Here, $a =12$ and $b =15$ .
$\text{Average rate of change} = \frac{k(15) - k(12)}{15 -12}$

STEP 4

Now we need to find the values of $k(15)$ and $k(12)$ . We can do this by substituting $15$ and $12$ into our function $k(x) = -16 \sqrt{x}$ .
First, let's find $k(15)$ $k(15) = -16 \sqrt{15}$

STEP 5

Next, let's find $k(12)$ $k(12) = -16 \sqrt{12}$

STEP 6

Now that we have the values for $k(15)$ and $k(12)$ , we can substitute these back into our formula for the average rate of change $\text{Average rate of change} = \frac{-16 \sqrt{15} - (-16 \sqrt{12})}{15 -12}$

STEP 7

implify the expression in the numerator $\text{Average rate of change} = \frac{-16 \sqrt{15} +16 \sqrt{12}}{3}$

STEP 8

Factor out the common factor of $16$ in the numerator $\text{Average rate of change} = \frac{16(-\sqrt{15} + \sqrt{12})}{3}$

STEP 9

Divide $16$ by $3$ to get the final expression for the average rate of change $\text{Average rate of change} =16/3(-\sqrt{15} + \sqrt{12})$ So, the average rate of change of $k(x)=-16 \sqrt{x}$ over the interval $[12,15]$ is $16/3(-\sqrt{15} + \sqrt{12})$ .

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