Final Answer:
The amount of paint in cubic centimeters needed to apply a coat of paint 0.050000 cm thick to a hemispherical dome with a diameter of 75.000 meters is approximately 1.77 x 10⁹ cubic centimeters.
Step-by-step explanation:
To estimate the amount of paint needed, we can use linear approximation. First, we need to find the radius of the hemispherical dome, which is half the diameter, so it’s 37.500 meters. Then, we can use the formula for the surface area of a hemisphere, which is A = 2πr², where r is the radius. Plugging in the values, we get A ≈ 2π(37.500)² = 17593.81 square meters. Since we want to find the volume of paint needed for a coat 0.050000 cm thick, we can multiply the surface area by the thickness to get the volume: V = A thickness = 17593.81 0.050000 = 879.69 cubic meters.
To convert this to cubic centimeters, we know that 1 cubic meter is equal to 1,000,000 cubic centimeters, so we multiply the volume by 1,000,000: V ≈ 879.69 * 1,000,000 = 8.7969 x 10^8 cubic centimeters. Rounding this to two significant figures gives us approximately 1.77 x 10⁹cubic centimeters of paint needed.
In conclusion, using linear approximation, we estimate that approximately 1.77 x 10⁹cubic centimeters of paint are needed to apply a coat of paint 0.050000 cm thick to a hemispherical dome with a diameter of 75.000 meters.