#### To determine

**To estimate:** The amount of paint needed to apply the coat of paint 0.05 cm thick to hemispherical dome with diameter 50m.

#### Answer

The amount of paint needed to apply a hemispherical dome is 1.96 m3.

#### Explanation

**Calculation:**

The volume of hemisphere with radius *r* is, V=23πr3 .

If apply the coat of paint to the hemispherical dome, then the radius of the dome is increasing and the volume of the dome is also increasing.

Thus, the extra amount of the volume is needed to apply the amount of the point.

That is, to find the change of the volume ΔV.

Here, the amount of paint needed to apply the coat of the paint is 0.005 m. That is, dr=0.005 m.

The differential is dV=f′(r)dr,

The derivative of the function f(r)=23πr3 is computed as follows,

f′(r)=ddr(23πr3)=2π3ddr(r3)=2π3(3r2)=2πr2

Substitute f′(r)=2πr2 in dV=f′(r)dr,

dV=2πr2dr

Substitute r=25 m and dr=0.005 m ,

dV=2π(25)2(0.0005)=2π(625)(0.0005)=2π(0.3125)=1.96

Since dV≈ΔV , the change the volume is 1.96 m3.

Therefore, the amount of paint needed to apply a hemispherical dome is 1.96 m3.