#### To determine

**To find:** The rate of change of height of the water inside the basin when it is half full.

#### Answer

The rate of change of the height of the water is dhdt=0.804 cm/min.

#### Explanation

**Given:**

A faucet is filling a hemisphere basin of diameter 60 cm.

The basin is filling with the water at a rate of 2 L/min=2000 mm3/min.

**Formula used:**

(1) Chain rule: dydx=dydu⋅dudx

(2) Volume of the hemisphere of radius r is:V=23πr3

(3) Volume of the portion of sphere of radius r from the bottom and height h is:V=π(rh2−13h3)

**Calculation:**

Let us assume that V be the volume of the hemisphere of radius r. And h be the height of the water inside the hemisphere.

The volume of the hemisphere of radius r=30 cm is,

V=23π(30)3=23π×27000=18000π cm3

Since the basin is half full, then

V=π(rh2−13h3)π(30×h2−13h3)=9000π(30×h2−13h3)=90003×30h2−h33=9000

On further simplification the cubic equation in h is as follows.

3×30h2−h33=900090h2−h3=27000(h3−90h2)=−27000h3−90h2+27000=0

From the numerical roots finder the three solutions of the above equations are as follows.

h1=19.58 ,h2=86.38 , and h3=−15.96 .

But out these three values

h2=86.38 is greater than radius of the hemisphere, which is not possible.

h3=−15.96 is negative and distance cannot be negative, therefore it is not possible value.

And the only possible value is h1=19.58.

Since the volume and height of the water inside the basin depend on the time t

Therefore, the volume and height are the function of the time t .

Differentiate V with respect to the time t .

ddt[V]=ddt[π(30h2−13h3)]dVdt=π[ddh(30h2−13h3)]⋅dhdtdVdt=π[30×2h−13×3h2]⋅dhdtdVdt=π[60h−h2]⋅dhdt

On further simplification the rate of change of height of the water is as follows.

dhdt=1π(60h−h2)⋅dVdt

Substitutes h=19.58 and dVdt=2000 in dhdt .

dhdt=2000π[60×19.58−(19.58)2]=2000π[1174.8−383.38]=2000π[791.42]=20002485.07 cm/min

On further simplification the rate of change of height is as follows.

dhdt=0.804 cm/min

Therefore, the rate of change of the height of the water is dhdt=0.804 cm/min.