Use linear approximation to estimate the amount of paint in cubic centimeters needed to apply a coat of paint 0.01cm thick to a hemispherical dome with a radius of 18 meters.
Hemisphere = 1/2 sphere
V = (2/3)πr^3
dV/dr = 2πr^2
dV = 2πr^2 dr
∆V = 2πr^2 ∆r
∆V = 2π(18^2)(.0001)
= .0648π
or
appr .2036 m^3
To estimate the amount of paint needed using linear approximation, we can consider the hemispherical dome as a flat surface. The linear approximation assumes that the change in the function, in this case, the volume of the dome, over a small interval is roughly equal to the slope of the tangent line at a certain point.
To start, let's find the volume of the entire hemisphere. The formula for the volume of a hemisphere is (2/3)πr^3, where r is the radius. Plugging in the radius of 18 meters, we can calculate the volume of the hemisphere:
V = (2/3) * π * (18)^3 = 24418.51 m^3.
Since we want to calculate the amount of paint in cubic centimeters, we need to convert the volume from meters to centimeters. Since 1 meter is equal to 100 centimeters, we need to multiply the volume by 1,000,000 (100^3) to convert from meters to cubic centimeters:
V = 24418.51 m^3 * 1,000,000 cm^3/m^3 = 24,418,510,000 cm^3.
Next, we need to estimate the change in volume due to a coat of paint 0.01 cm thick. Since the thickness is relatively small, we can assume that the change in volume would be proportional to the change in thickness.
To estimate the change in volume, we can use the formula for a thin cylindrical shell:
dV ≈ 2πrh * dt,
where dV is the change in volume, r is the radius of the hemisphere, h is the thickness of the paint coat, and dt is the small change in thickness. In this case, r = 18 meters, h = 0.01 cm, and dt = 0.01 cm.
Plugging in the values, we have:
dV ≈ 2π * 18m * 0.01cm * 0.01cm = 0.01136663 m^3.
Again, we need to convert the change in volume from meters to cubic centimeters, so we multiply by 1,000,000:
dV ≈ 0.01136663 m^3 * 1,000,000 cm^3/m^3 = 11,366.63 cm^3.
This estimated change in volume represents the amount of paint needed to cover a small portion of the hemisphere's surface with a coat of thickness 0.01 cm.
Finally, to estimate the total amount of paint needed to cover the entire dome, we divide the total volume of the hemisphere by the change in volume per coat:
Total coats = V / dV = 24,418,510,000 cm^3 / 11,366.63 cm^3 ≈ 2,150,691.47 coats.
Therefore, we estimate that approximately 2,150,691.47 coats of paint, each 0.01 cm thick, are needed to cover the hemispherical dome with a radius of 18 meters.