Final answer:
To estimate the paint volume for a hemispherical dome using linear approximation, calculate the surface area (2πr²) with radius converted to centimeters, and multiply it by the paint thickness (0.08 cm), resulting in 6168000π cm³.
Step-by-step explanation:
To estimate the amount of paint needed to apply a coat of paint 0.08 cm thick to a hemispherical dome using linear approximation, we need to calculate the surface area of the dome and then multiply it by the thickness of the paint layer.
The formula for the surface area (S) of a hemisphere is S = 2πr², where r is the radius of the hemisphere. Given that the diameter of the dome is 70 meters, the radius, r, would be half of that, which is 35 meters (or 3500 cm, since we need to work in centimeters to match the thickness of the paint layer).
Using the surface area formula:
- Surface Area = 2π(3500 cm)²
- Surface Area = 2π(12250000 cm²)
- Surface Area = 77100000π cm²
Now, we multiply the surface area by the thickness of the paint layer:
- Volume of paint = Surface Area × Thickness
- Volume of paint = 77100000π cm² × 0.08 cm
- Volume of paint = (77100000π cm²) × 0.08 cm
- Volume of paint = 6168000π cm³
This gives us the estimated volume of paint needed in cubic centimeters.