Use linear approximation to estimate the amount of paint in cubic centimeters needed to apply a coat of paint 0.040000 cm thick to a hemispherical dome with a diameter of 60.000 meters

Volume sphere= 4/3PI r^3

Dv/dr= 4PI r^2
dv= 4PI r^2 dr= 4PI*900*.0004 m^3. That is a lot of paint. check my thinking

To estimate the amount of paint needed to apply a coat of paint to a hemispherical dome, we can use linear approximation.

First, let's calculate the surface area of the hemispherical dome. The surface area of a hemisphere is given by the formula:

A = 2πr^2

Given that the diameter of the dome is 60.000 meters, the radius (r) is half of the diameter, which is 30.000 meters.

Now, let's calculate the surface area of the dome:

A = 2π(30.000)^2
= 1800π square meters

The next step is to convert the thickness of the paint into meters, as the dimensions need to be consistent. The thickness of the paint is given as 0.040000 cm, which is equivalent to 0.000400 meters.

Now we can use linear approximation. The linear approximation formula is given by:

ΔV = A * Δd

Where:
ΔV is the change in volume (amount of paint)
A is the surface area of the dome
Δd is the change in thickness (0.000400 meters)

Therefore, the change in volume is:

ΔV = (1800π) * (0.000400)
≈ 2.26274 cubic meters

Finally, we convert the volume from cubic meters to cubic centimeters since the question asks for the amount of paint in cubic centimeters:

2.26274 cubic meters * (1000000 cubic centimeters/1 cubic meter)
≈ 2,262,740 cubic centimeters

Therefore, approximately 2,262,740 cubic centimeters of paint are needed to apply a coat of paint 0.040000 cm thick to the hemispherical dome.