Respuesta :
Answer:
(i)=263.9 micrometer square
(ii)=155 micrometer square
(ii)=116.9 micrometer square
(b).37.7 micrometer square
Step-by-step explanation:
Given,
[tex]f\left ( x \right )=V=4/3\Pi r^{3}[/tex]
Average rate of change V with respect to a to b where a and b are radius
[tex]\frac{\mathrm{d} V}{\mathrm{d} r} =(f\left ( b \right )-f\left ( a \right ))/\left ( b-a \right )[/tex]
(i) 3 to 6 micrometer
[tex]f\left ( 3 \right )=4/3\Pi \times 3^{3}=36\Pi[/tex] [tex]f\left ( 6 \right )=4/3\Pi \times 6^{3}=288\Pi[/tex]
[tex]\frac{\mathrm{d} V}{\mathrm{d} r}=\left ( f\left ( 6 \right )-f\left ( 3 \right ) \right )/\left ( 6-3 \right )[/tex] [tex]=(288\Pi -36\Pi)/3[/tex]
[tex]\frac{\mathrm{d} V}{\mathrm{d} r}=84\Pi =263.893[/tex] [tex]=263.9[/tex] micrometer square
(ii) 3 to 4
[tex]f\left ( 4 \right )=4/3\Pi 4^{3}=256\Pi /3[/tex]
[tex]\frac{\mathrm{d}V }{\mathrm{d} r}=\left ( f\left ( 4 \right )-f\left ( 3 \right ) \right )/\left ( 4-3 \right )[/tex]
[tex]\frac{\mathrm{d} V}{\mathrm{d} r}=(256\Pi /3-36\Pi)/1=154.985[/tex]
[tex]\frac{\mathrm{d} V}{\mathrm{d} r} =155[/tex] micrometer square
(iii) 3 to 3.1
[tex]f\left ( 3.1 \right )=4/3\Pi 3.1^{3}=124.788[/tex]
[tex]\frac{\mathrm{d} V}{\mathrm{d} r}=\left ( f\left ( 3.1 \right )-f\left ( 3 \right ) \right )/\left ( 3.1-3 \right )[/tex]
[tex]=\left ( 124.7882-36\Pi \right )/\left ( 3.1-3 \right )[/tex]
[tex]\frac{\mathrm{d} V}{\mathrm{d} r}=116.9091[/tex]
[tex]\frac{\mathrm{d} V}{\mathrm{d} r}=116.9[/tex] micrometer square
(b) At r=3 micrometer
Instantaneous rate
[tex]\frac{\mathrm{d} V}{\mathrm{d} r}=\frac{\mathrm{d} (4/3\Pi \times r^{3})}{\mathrm{d} r}[/tex]
[tex]\frac{\mathrm{d} V}{\mathrm{d} r}=4/3\Pi \times 3r^{3-1}[/tex]
[tex]\frac{\mathrm{d} V}{\mathrm{d} r} =4/3\Pi r^{2}[/tex]
[tex]=4/3\Pi \times 3^{2}=37.699[/tex] micrometer square
[tex]=37.7[/tex] micrometer square
