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(a)

Volume of the growing sphere cell is , Where  is a radius and is measured in micrometers.

 micrometer meters.

(i)

Average rate of change if radius  changes from  to  :

Average rate of change of  with respect to  is defined as  on the interval .

Here  ,   and .

Average rate of change in volume is .

 m².

(ii)

Average rate of change if radius  changes from  to  :

Average rate of change in volume is .

 m².

(iii)

Average rate of change if radius  changes from  to   :

Average rate of change in volume is .

 m².

(b)

Instantaneous rate of change when   :

Instantaneous rate of change of  with respect to  when  is defined as .

Instantaneous rate of change of area at  is .

.

Differentiate on each side with respect to .

.

.

(c)

The surface area of the sphere is  of the circle is , where  is the radius of the sphere.

Consider .

Therefore rate of change of volume of sphere with respect to its radius is equal to the its surface area.

If the radius is increased by an amount of .

Geometrical view of  with an increased radius .

Therefore volume of sphere with increased radius of  is .

If  is small, then we can approximate .

Rate of change of area of new sphere  with respect to radius can be considered as

 The rate of change of volume with respect to its radius is equal to the its surface area.

(a)

(i) Average rate of change if radius  changes from  to  is .

(ii) Average rate of change if radius  changes from  to  is .

(iii) Average rate of change if radius  changes from  to  is .

(b) Instantaneous rate of change when   is .

(c) The rate of change of area with respect to its radius is equal to the its circumference.