Question

unit 2: reviewengagementobjective:create an exponential function to model a relationship between quantities in order to solve mathematical and real-world problems.standard:fbf.1, fbf.1a, fif.4, fif.6, fif.7, fif.7e, fle.2, fle.5warm up. show your worka company manufactures spherical ball bearings. the function $c(r)=50(\frac{4}{3}\pi r^{3})$ represents the total volume of 50 ball bearings with a radius of r millimeters. the table below shows several values for r and c(r) rounded to the nearest hundredth. calculate the average rate of change in the total volume as the radius increases from 2 mm to 8 mm$m=\frac{y_2-y_1}{x_2-x_1}=\frac{f(b)-f(a)}{b-a}=\frac{f(8)-f(2)}{8-2}LXB0v = \frac{107,233.03 - 1,675.52}{8-2}$| radius r (mm) | total volume c(r)($mm^{3}$) || ---- | ---- || 2 | 1,675.52 $f(a)$ || 4 | 13,404.13 || 6 | 45,238.93 || 8 | 107,233.03 $f(b)$ |students, draw anywhere on this slide!pear deck interactive slidedo not remove this

Explanation:

Step1: Recall average rate formula

The average rate of change of a function $C(r)$ from $r=a$ to $r=b$ is $\frac{C(b)-C(a)}{b-a}$

Step2: Identify values from table

For $a=2$, $C(a)=1,675.52$; for $b=8$, $C(b)=107,233.03$

Step3: Substitute into formula

$\text{Average rate} = \frac{107233.03 - 1675.52}{8 - 2}$

Step4: Calculate numerator and denominator

Numerator: $107233.03 - 1675.52 = 105557.51$
Denominator: $8-2=6$

Step5: Compute final value

$\text{Average rate} = \frac{105557.51}{6} \approx 17592.92$

Answer:

The average rate of change is approximately $17592.92$ mm³ per mm.